as much as I wish my book was number one on all of Amazon it is number one in the humor section right now so I will take that it's called like living among scorpions and I notice it is three o'clock now what did Father Gately tell us we're supposed to do let's say it all together now here the three o'clock hour Jesus I trust in you I can't think of a better way to begin a talk so thank you so much for joining me in that it I love this conference I am so excited about this conference in particular because it is so focused on the truth finding objective truth not just how we feel good about what we believe not just our experiences but why do we believe what we believe that's that is something that really appeals to my little nerd's heart as you'll see in my talk my dad very much raised me to be a truth-seeking person you know he would ask me questions like when I would talk about how neat it was that people have landed on the moon my dad is the type of guy who would say well why do you believe that how do you know some people say it was a hoax how do you know it wasn't a hoax of course he didn't think the moon landing was a host but a hoax but he wanted to give me that kind of training he very much instilled in me this desire to know what is true to ask tough questions to question assumptions and to always seek the truth even if it is inconvenient and I always warn atheists be careful about that because your kids will grow up to be Catholic if you raise them that way so I can't wait to get into that story but I think what you'll see as you hear my story is that is it is the story of someone who was obsessed with knowing the truth finally found it but found that there was one thing she did not understand about the truth and it's the most important thing that you could ever know so I can't wait to share all that with you let's get started let's start at the beginning I never believed in God when I was younger not once not even when I was five years old I didn't wonder if there might be a God I didn't feel anything in my heart a strict atheist materialist worldview just seemed to me like it was obviously true and I very much inherited this way of thinking from my dad my dad is an engineer his father was an engineer got that great rigidly logical way of thinking now my and he was an atheist he he is well actually he says he's agnostic now so pray we're we're moving in the right direction he's the kindest man ever takes my kids to all their activities he's a good-hearted man so please say a prayer for him but at the time he was a staunch atheist he had done a lot of research in his own life he did not get the answers to the questions he was looking for about belief he didn't have access to wonderful conferences like this one so he became an atheist but he always said that he was not raising me to be an atheist per se as much as he was raising me to seek truth in question assumptions seek truth no matter what even if it is inconvenient for you that was kind of a motto of my upbringing now my mother was not an atheist she is what philosophers call a Macy's and that is someone who is more concerned with whether Macy's has an additional 30% off this weekend then whether there's a compelling case for a monotheistic God my mother is wonderful also just super helpful super generous and she's one of those people who kind of feels like well I'm gonna try to be a good person I'm gonna try to do the right thing and and I think she almost sees questions of theology as as not relevant as long as you're trying to do the right thing so I always tell her you know you're on the right path just keep you know keep following that line of thought so I it just never occurred to me to even wonder if there might be something else out there I was very much in this mindset of you only believe things for which you can find evidence it seemed dangerous to me to believe in things for which there's not evidence I mean I know this podium exists because I can I can feel it I can see it I could weigh it measure it but the idea of believing in something that was just kind of out there that you couldn't experience with any of the five senses it just kind of seemed like well how would you even know that was true how would you know if you weren't part of some church that was manipulating you in into believing falsehoods so all my life I was an atheist in and I really came to embrace my belief system in college in college I started out at Texas A&M you might have heard of that university is there an Aggie in here did I hear a whoop oh my gosh we've got some Aggies all right now now Texas A&M it is like the Steubenville of Texas full of Christians full of Catholics bad place to be if you are a militant atheist and the Christians down there were doing their job well because I transferred it was just too religious I could not take the religious talk anymore so I ended up at the University of Texas in Austin and I looked around and I said ah atheists yeah this this is more like it this is the culture I'm looking for so being in that kind of environment really helped me embrace my faith or lack thereof and looking back I am so glad you thank you Jesus that this was just before the internet really broke and people could have blogs and put their opinions out publicly because my understanding of my own atheistic world views and worldview and my friend's understanding oh it he was it was just embarrassing some of the things we would sit around and say I mean I real I remember having discussions like saying oh those religious people are so stupid what about suffering why would a loving God allow suffering that they never thought of that one I mean it was so embarrassing we were asking the most basic questions and seriously thought that in 2000 years we were the first people to come up to this as 19 year olds who knew everything and in a perfect way to summarize this is if you ever want a laugh if you ever want to read just the most fabulous brilliant little essay you need to look up the London review of books review of Richard Dawkins God at delusion now you know that book that's the book that your 17 year old atheist nephew keeps telling you you need to read it's something of a of an atheist Bible and so many people say that oh it just it just demolishes religion oh he just he just nails it and the argue I've read the book it's it's unbelievable I thought that I would do a series on my blog showing where he was wrong but in every pair every paragraph needed to be addressed he confused Protestant Christianity with Catholic beliefs oh I mean to use the popular phrase the thing was a hot mess and that that is modern atheism in a nutshell I mean throughout history there have been some atheists that had some arguments that are worth dialoguing about but the new atheists have none of them so anyway the London review of books they get a hold of the God delusion opening line first thing they say so my favorite quotes ever they say imagine someone holding forth on biology whose only knowledge of the subject is the book of British birds and you have some idea of what it feels like to read Dawkins on theology absolutely me I was no better than Dawkins I had all these half-baked ideas and it could see I was the product of this education system of the modern education system which teaches children facts but it doesn't teach them how to think it doesn't teach them it doesn't help them distinguish between the different disciplines it doesn't tell them things like if you want to know the approximate number of galaxies in the universe that is a question for science but if you want to know why the universe is here at all if you want to know why something exists instead of nothing that is not a question for science that is a question for philosophy but me and all of my new atheist friends we had no concept of that and we were seriously steeped in this culture where we honestly thought that the scientific method is how you know everything if you can't prove it materially objectively through the scientific method it must not be true we were a very lost generation so to fast forward a bit I started to work as a programmer at a high-tech company and that is where I met my husband my husband is a really smart guy he grew up poor so poor in fact that I mean his mother often didn't know how she was going to put food on the table they very often they couldn't even run the heat in the winter well now it was Houston so it wasn't that bad but but he was very poor and she had heard of these places like Harvard and Yale and no one in her family had gone to college I mean she didn't knew much about it but she she had heard that if you go to places like that you won't be poor and that is all she knew she didn't even know where they were but she knew they existed somewhere and that if you go to places like that you won't be poor and so this is the type of guy my husband is she told him to go to schools like that and he said okay and he ended up going to Yale undergrad graduating in three years Columbia Law School Stanford Business School and studied in the master's computer science program at Stanford while he was there because why not and he had this amazing network of people he was actually he had an opportunity a possible opportunity that if it had panned out he would have been employee number 16 at a little company out there in Silicon Valley called Google I but what but if he had taken that job he would not have moved to Austin which is where we met and I always tell him honey who needs billions of dollars when you have me and depending on the day it's more or less a comforting thought so so he's obviously a really smart guy so we we met at a high-tech company and and so to fast forward a bit so we got married and one of the things I learned it we had we had a completely secular ceremony I was really into this idea that we lived in a post-Christian culture we weren't gonna have anything related to Christianity at our wedding and we rented a theater we weren't even married at our own wedding cuz like I had to make a statement about how the state can't tell me if I married married or not it was very complicated and you know here's a little tip if you are ever thinking about reinventing the wheel on a thousands of year old tradition you might want to rehearse it first we were too busy planning the after-party that ended up lasting for 12 hours so it's good that we spent so much time planning it we didn't rehearse our own ceremony and Joe he knows people from everywhere he had people fly in from all over the world but our ceremony because we wrote our own vows and we didn't rehearse it it only lasted seven minutes we had people from London from Africa and they were thinking we flew across the world for this and I wore a dark purple dress and as I was walking out one of my one of my Texan relatives and his cowboy hat said where's the bride disaster so is again the lesson is if you are going to reinvent the wheel on a thousands of year old tradition just make sure you rehearse it first so so we got married and a little over a year later we had our first child and I hadn't really been thinking of questions of theology at this point about atheism about anything like that and I I I'd kind of put it to the side and this is something this is kind of an arc that you see and a lot of young atheists they're really into being atheists when they're in college and then they kind of get a job and they just you know they don't care as much anymore so those I had just really gotten into this blasé attitude about anything having to do with faith but then it all came bubbling up again when my first child was born because you know it's one thing to mess up your own life it's one thing to make mistakes in your own life but when you have a kid you really want to get it right and it occurred to me that you know I'd had some questions about the atheistic worldview that I had never really explored and I was still pretty sure it was right but I started to think that maybe I should just make sure it's right before I pass this along to my own child and then and and then just the experience of being a mother had such an impact on me you know something to understand is I grew up in very secular circles where you know babies were just not a part of the mix I mean I never had a friend growing up never who had a baby in the house I don't think I ever had a friend who had more than two siblings certainly no more than two siblings living at home so I had kind of I mean I was so disconnected from that side of the human experience I mean I think in the back of my mind I didn't know where new people come from I kind of thought that like they all come from from a cloning room at the back of Starbucks and they walk out with their wire rim glasses in their latte so I mean having a baby was like whoa I was really kind of knocked off kilter by that and so I looked at this child when he was about two weeks old and I and I thought what is he in a strict atheist materialist worldview not borrowing from spirituality not borrowing from Judeo-Christian tradition not what I wanted him to be what is he truly in in the belief system that I claim to profess that's something that my dad always trained me and he says follow your beliefs to all their logical conclusions don't flinch really think through them so I looked at this baby and asked what is he in the atheist materialist worldview and I looked down at him and I said oh he is the most precious little randomly evolved set of chemical reactions that came from nothing and will return to nothing I was like this is just false I don't believe that this is true the love that I felt for that child was so real and it wasn't that I just wanted it to be real that this was a mother trying to impute more value than it was actually there I knew that the love that I felt for that child came from some external source to the chemical reactions in my brain was real in the sense that if the whole world blew up tomorrow that love would be there and would still exist in some sort of eternal way I knew that more than I had ever known anything in my life and that was the first moment that I was no longer an atheist so I knew what I didn't think was true I didn't think that the atheist materialist worldview was true the problem is that I didn't know what I did think was true so I thought okay so I'm not an atheist anymore so what does that mean and I thought well okay so I guess I believe in some sort of spirituality but you know I didn't want to take that too far so I thought I thought I would do something crazy and say a prayer now I'd seen this in the movies and you know I went to Texas A&M for a while I mean I'd seen people pray I was aware that this is the thing that is done but I had seriously I'm in my late 20s at this point I had seriously never said a prayer in my life and I didn't I had all these questions these really nerdy analytical questions like well how do you go how do you know when you go from talking to yourself to talking to whatever might be out there I mean is there like do you go like is this thing on testing testing I mean I don't I really didn't know and so I said the lamest prayer anyone has ever said I just basically said if anyone or anything is out there things plural I'm in Austin I want to be open-minded here I'm open to hearing from you now and then I said amen because I'd seen it on TV so nothing happened my room was still silent there was no mysterious knock at the door with a message I mean look I had seen the Hallmark movies I had even occasionally seen some like Christian specials and if I knew one thing about religion I knew for sure that the moment you first say a prayer things start happening I mean frankly I would have expected a chorus of angels some lights bare minimum though crazy coincidence you know like someone walks up and hands me a Bible which would have been weird because I was in my bedroom and it was like two o'clock in the morning but still hey this is prayer it's supposed to make amazing things happen so I just said well there you go God doesn't exist so that's that was easy that'll move me on to the next phase because obviously if a loving personal God exists and wants me to know him he would have responded in some way so I looked at it like well I I tried to call and it went to voicemail so to speak so I can check that off the list does does a personal God exists who cares if I know him no so moving on so I live in Austin so of course I started dabbling in Buddhism you're required by law to dabble in Buddhism for at least three months in my city so I you know I started reading those books and then it's totally out of the blue in one of the crazier coincidences of my life I was in a bookstore and I happened to come across a Christian book now this was unlikely for a wide variety of reasons one of them which is I had never been in the religion section let alone a Christianity section of a bookstore or library in my whole life with one exception when I was in the fourth grade I took all the Bibles from the religion section and I moved them to the fiction section in what was the edgiest prank the entire fourth grade has had ever seen I was convinced so just so randomly I come across this book the case for Christ by Lee Strobel and I couldn't put the thing down now wait I need to jump in and say something here usually when I name that book and people know I was lifelong atheists converted to Catholicism they've got out there smartphones they've got it on one day shipping to their atheist son who's in college it's important to note notice I'd said that prayer it was a really lame prayer but I had said a prayer a few months before I could have come across this exact same book and I would have laughed at it thrown it aside just a few months before so I was in a state of openness nobody will ever encounter the truth if they're not open to it so it's important as I say what's next that you understand that I was in a state of openness so I'm reading through this book and it really caught my attention but I'm still in the bookstore and I had to go and I thought I can't I can't buy a Christian book what if someone sees me I mean I know people in this town that's I'm kind of like hiding it behind my back in case anyone I know is here and and I walked away and then I walked back and said I have to get it so I go up to the checkout and and then and other people started coming up and I'm trying to get this book and so I knew what I would do I loudly said to the cashier I said I'll need a gift receipt for that because that is a gift I have a friend she's a Bible person I don't know gift receipt in that bag I need it really need it so I take this book home and I read it cover to cover and it wasn't that it was perfect there were some chapters at least trouble wonderful guy he's a Protestant pastor even now there are some chapters like especially about authority and issues that you know they're not perfect but he said at the beginning of this book he said my goal is not to convert you to Christianity on the spot my goal is simply to get you to see that this is worth looking into that there is enough of a historical case for Jesus Christ being who he said he was that you should explore it further and I'll give you just one example I mean there are a lot of things that I found compelling about it but to just give you one example so at least trouble was interviewing a philosopher with a background in chemistry named JP Moreland and Moreland is talking about the the religious and social institutions that have been so important to the Jews over history and Moreland says he points out that you know these Jews believed that to abandon these social structures such as keeping the Sabbath holy animal sacrifice would risk their souls being damned to hell after death and this is Moreland talking now a rabbi named Jesus appears from a lower class region he teaches for three years gathers a following of lower and middle class people gets in trouble with the authorities and gets crucified along with 30,000 other Jewish men who are executed during this time period Moreland goes on to say but five weeks after he's crucified over 10,000 Jews are following him and claiming that he is the initiate the initiator of a new religion and get this they're willing to give up or alter all five of the social institutions that they have been taught since childhood have such in have such importance both sociologically and theologically something huge happened in first century Palestine and Lee Strobel pointed out when you when you look at how much these people were willing to sacrifice and how many of them were for this new religion I mean becoming a Christian meant persecution Lee Strobel pointed out that you know people might be willing to die or suffer for things that they think are true but they're incorrect about but you have all these people but nobody is willing to suffer and die for something that they know is false and yet you have all of these people saying that they are following Jesus Christ and that he is the Christ and that he is the founder of this new religion and they are willing to suffer and die for this and in Strobel said no one would be all these people who claim to see Christ risen from the dead would not have been willing to do that if they were lying and so he takes a step back and says you have to admit that something happened what happened here and he proposed a really interesting theory maybe they really did see Christ risen from the dead so that got me thinking I was certainly not anywhere near ready to convert to Christianity I had all of these negative stereotypes about this religion but it did got me thinking and so I decided to go get a Bible I'd never owned a Bible before I knew nothing about the Bible I was as sheltered from this book as you could possibly be and live in America today and it caused me no end of embarrassment when we moved around a lot I moved I lived in the Bible Belt many times in different places and sometimes my parents would let me go to Sunday school with little friends not even though they didn't agree with the religion they were not going to get up early in the morning after I'd done a Saturday night spend the night so I ended up at Sunday school and I remember one time that I always knew to use the line that when they asked where I went to church I would say something vague like oh it's a it's a country sure yeah you wouldn't have heard of it it's but I do definitely go to church and one time the sweet Sunday school teacher took all the books of the Bible and did a game where you can put the books of the Bible in order and she said why don't we let our new visitors start oh I had no idea but I thought I could have I thought I could make a pretty good guess like numbers would be in the beginning because that was probably a table of contents now I had heard of the names like Matthew Mark Luke and John every author knows that you start with your strongest character at the beginning so I went from numbers Matthew Mark Luke now there's some big words I didn't know so I put them at the end revelation would probably be first because you know you kind of want to start with a bang so she looks at this list of the books of the Bible and she leans down and she said honey where did you say y'all's family goes to church again I knew nothing about the Bible and even in college I intentionally avoided courses that were that would talk about the Bible even from a historical perspective because I was into this idea that we live in a post-Christian culture and it's irrelevant so I skipped all that so I get my first Bible and you know I'm a writer I like to write and I know good writing skills so I thought that you know I'd always heard that you should introduce your main character no more than 20% of the way into the book that's a big writing rule now Jesus he's the main character of the Bible I'm expecting him to come in about page 30 so I'm reading through it I had also through these Bible school moments I had picked up through cultural osmosis that there was something about their fishermen and they sang songs and there was bread and wine that was also in the Bible so I am in Deuteronomy like whoa whoa whoa and then like in thousand the army was wiped out and I'm like where are the fishermen this is crazy and I was driving Joe crazy because it Joe would come home stressed out from work and I win and he'd say oh he has some problems and I would say Joe have you tried sacrificing a bird over a clay pot and dipping its blood and yarn and hyssop it's in the Bible I'm reading the Bible I think I know so Joe took me aside and said stopped New Testament that's the New Testament and so and so oh that's what I was missing and now I'm gonna you know by this time I'd really I felt something with this Jesus person I felt like I was getting close to something real and so I opened up the New Testament it was a list of names like just a bunch of names and like and then you get to the end and it's like some guy's dream and then I was waiting I wanted a bulleted list of like here are the reasons why we believe what we believe and there should at least be a strong call to action at the end like if you are convinced by this testimony then here's what you do next and it's like some vision like I don't know what to do with this I was I could not have been more confused and and so then so I thought okay I'm gonna stick with this though so I'm reading the New Testament I did not see a clear case for what I knew to be the traditional Christian moral code abortion euthanasia the death penalty I just I just didn't see this spelled out anywhere and so of course I turned to the internet like a good nerd and I and I would ask people on the internet like why I'm just I am not seeing this and they would say well you know you need to read the New Ted the Old the New Testament with the entire New Old Testament so that you know how to interpret the scriptures in concert with one another and then it also helps if you have a biblical encyclopedia and a biblical dictionary some reference materials and I wanted to say don't you guys have jobs because like I have a little kid here and my husband works all the time and like when is it that you think that I have time to become like a PhD in sacred scripture like I this is this really the system that God wanted to hand us and by the way guys this is not a system that works without the printing press none of the biblical in in concordance and encyclopedia those are all great impossible for the average person to have before the printing press I think I once read that it took monks something like two thousand hours to hand copy a copy of the Bible they were very expensive and so I said you know the printing press was invented in the 16th century but how on earth could people ever come to know Christ before that especially that they were out working in the fields you know they didn't have time to read the Old Testament and the New Testament but the biggest thing that concerned me is when I would ask these Christians what they believed everyone like what the moral code is every single person seemed to give me a different answer but they all had scripture to back up their case and so some some Christians would say well we don't like abortion but it is necessary sometimes and euthanasia is okay as long as it's compassionate death penalty you know I could gonna go either way everyone had scripture to back up their claims so here was the problem as someone who is totally unfamiliar with this religion what I wanted to say to the Christians was I need a very fundamental question answered here and that is the question of who is Jesus Christ you guys are telling me that I should be part of this religion and so I need to know who is it that you want me to follow because let me tell you the Jesus Christ over here who says hey man whatever y'all want to do is cool marriage whatever however you want to define that that's cool abortion contraception death penalty I you know really just as long as you feel positive things in your heart I'm down with it that is a different Jesus than the Jesus over here who calls us to the very highest standard of love for our human for our fellow human beings who says that marriage means a very specific thing because it is the protector of new human life and it is the protector of human love who says that we must respect all human life from the moment of conception until natural death that is a very very different Jesus than the happy-go-lucky Jesus over here and I could not get a clear answer from modern American Christianity about which Jesus I was supposed to believe in now meanwhile I had started a blog because that is what you do when you're a huge nerd and you're having an existential crisis I did not have a lot of Christian friends oddly enough couple decades of being a loudmouth militant atheist does not leave you with a wide social network of devout believers so I thought maybe I could find some Christians on the internet kind of an interesting side note my blog when you start a blog you don't have any readers so I recruited people to come talk to me I went into some really heavy heavy atheist forums and and I would find people they were tons of Christians but I handpicked 10 or 15 who did an amazing job of just demolishing atheist arguments it was embarrassing for the atheists they knew more about history than the atheists they knew more about science than the atheists they knew more about the human experience and and again this was a small percentage of the Christians on there but I said these guys have something that's real and I want it so I emailed them individually I would find out later to a man to a man every single person I had identified as being able to demolish atheist arguments was Catholic but at this I didn't know that I didn't know that at this time because I just started the blog so I had those issues of not knowing who Jesus really is so I go back to the blog and I said guys I I hate to tell you this but I have got some bad news your religion's totally bunk I'm sorry it's you know I've been investigating it can't get a clear answer on your moral code can't get a clear answer on who Jesus is so you know there you no one knows how to interpret the Bible everyone it's like their own personal book that they like project their personality on to and say it's from God this is a disaster so there you go and I kind of thought they would be like well that's a great point does anyone know where the nearest Buddhist temple is can't argue with that but they came back and they said something really interesting they said what if before Jesus returned to heaven he founded a church one church that he instilled with his own authority what if he did with this church what he did with the writers of the Bible and that he took fallible imperfect people and used them to convey infallible perfect truth what if he promised to stay with this church to the end of the age so that it could speak truth to all cultures in all times in all places and I heard that and I said this yes this theory makes so much sense this explains how people could do Christianity before the printing press this explains how illiterate people could come to know God that was another question I was wondering when people were telling me about in biblical encyclopedias and concordances and reading the whole Bible I thought the world literacy rate is only 80% and that is much higher than it's ever been in history like how do illiterate people ever know who Jesus is so this explain this this idea of this living church it explains how illiterate people can come to know God people without free time to read the entire Bible this makes so much sense yes yes yes and they said that's the Catholic Church and I said no no no what what are you talking about I I can't become Catholic I mean obviously it's it's archaic it's patriarchal it oppresses women I mean you name a bad stereotype about Catholicism I had it so I said no no there's no way I could become Catholic and Joe he was so Joe was at the time that we got married Joe was kind of a lukewarm very much fallen away Southern Baptist and so he had all the anti-Catholic stereotypes and so he'd been kind of following along sort of amused with all of this and so he when he when he saw this turn of events I mean this was just I had just gone off the rails and he said you know it's bad enough that you've been messing around with this Catholic stuff and now you're messing around with these Catholic internet people you need to you need to take a step back but I said to him I said Joe I don't have anywhere left to turn because this is the only theory within Christianity that is even possible and I said I I think I am on to something with this Jesus person I can't explain it but I feel like there is something there and I have nowhere else to turn but the Catholic Church and somewhat reluctantly he agreed and he got on board with me and so we both began researching this we bought all the books we had stacks and stacks of books and it was kind of interesting to see how his questions as a Protestant differed from mine like he was he kept wrestling about the role of Mary that just made so much sense to me from the very beginning I I mean not that I wasn't sure if I believed all this but I said look let me tell you this Joe so God the creator of all the galaxies and all the stars and planets in the whole universe so if he if this is true that he came down to earth then that means he got to choose his own mom don't you think she'd be pretty special in the one true church maybe yeah Marian devotion boom easy but I had I had I had I had I had different issues that we had to work out but the big thing for us was the most controversial issues of the church abortion contraception when we realized that the church was right on those issues I mean we could not have been more shocked I was always militantly pro-choice I donated to Planned Parenthood I was as pro-choice as you could possibly be I didn't even know you could be pro-contraception I thought it was an urban legend that anyone was even anti it so so when I read the Catholic Catechism I was shocked that even the stuff that I was most certain that I would disagree with it just made sense and it was the most internally consistent belief system I had ever seen and when I read it when I found out what Catholics actually believe Joe and I both agree we had all these different perspectives but we both came together on one thing we looked at this and we said it's like we have discovered the owner's manual to the human soul whoever put this together whoever put this body of wisdom together knows us better than we know ourselves so this was the final thing I had been looking for all along I had been wondering as I was getting closer and closer to the church I thought well how would I know if it's true I mean this theory makes sense and it all sounds right but this is this is a big leap to become Catholic I mean how would I know if this church really were guided by God and what I had come to as I said I would need to see something here that humans cannot do on their own I would need to see the fingerprints of God somewhere and when I saw this moral code that was so reviled by the world and so counterintuitive yet was so perfect and it was like everything that had been written on the human heart just poured out and clarified and when I saw that this church has been teaching this same truth never reversing itself saying the exact same things for 2,000 years after empire after empire has fallen away around it I said here is something that humans cannot do on their own I had found the fingerprints of God so Joe and I both became Catholic at St. William and Round Rock Texas that's that is still our parish today we both became Catholic at Easter Vigil 2007 and that's where the story should end right you become Catholic and then you don't have problems anymore and life is easy like that's kind of what I expected frankly I mean seemed pretty cool like God's on my side now so we've been I know the truth so whoop boy I'm glad all that suffering is over with oh little did I know so right before we became Catholic I mean a few weeks before we became Catholic I got a deep vein thrombosis here in my right leg that is a blood clot in a major vein they are very dangerous it was in my second pregnancy it turns out that it was caused by a blood clotting disorder that makes pregnancy very dangerous for me it makes it likely that my that my blood will clot when I'm pregnant it is a rare disorder it is an inherited disorder and I inherited it from both parents and the hematologist when he diagnosed me he kept there was kind of this look on his face and I think he was trying to find a delicate way to ask did your parents meet at a family reunion or there's something you're not telling me I mean nobody has two copies of this gene so it's it's really crazy and it's again the big thing was that it made pregnancy just a mess so first of all treating this issue that I had it was enormously expensive we were self-employed had really bad insurance and then also just the stress of going through all that and so we were just kind of in a bad state we also lived with my mom Joe was starting a business it was just a really rough time and so then in my first lesson as a Catholic that God's timing is sometimes different than your own our third child was on the way just a few months after my second got here and this was I had barely healed from the DVT oh my doctors I thought they were gonna kill me with their bare hands I mean they they just they thought I they just thought it was so crazy but we did we were able to manage this condition with this blood thinner the trick is the blood thinner costs a thousand dollars a month and that is that's my insurance covered it that was my portion it is outrageously expensive I mean you could like buy I used to think of what what else I could buy for that money so we had been wiped out from the last pregnancy we still lived with my mom there weren't even there wasn't really even room for me and Joe to live with my mom so now bringing a third child into the mix it was it was a low time for me I was so stressed out about money we were on the brink of bankruptcy and I thought this is it so I guess Jesus was serious about all this stuff in the New Testament about how you're gonna suffer if you're a Christian oh this is lame I thought it was gonna be I didn't think he was serious about that because I read all this stuff but because you know the Lord had had really called us into this life of you know we used to live downtown and drive a jaguar and all that and the Lord seemed to be calling us away from that but I was saying this this is miserable this is hard I am not happy I do not like this this Catholic life thing so at that time I was kind of trying to pray in my own bumbling way and I just told the Lord I am miserable and this is too hard and I don't think I can do this frankly and so this quote came to me and it's one of those things I just I read it somewhere someone posted it on the internet or something and it was I don't know if you've ever had a moment like this but I just I knew that these words were what the Lord wanted me to hear right now it's from blessed Elizabeth Hesseblad and she says dear Lord I do not ask to seek the path in darkness in anguish in and fear I will hang on tightly to your hand and I will close my eyes so that you know how much trust I place in you spouse of my soul as I reread those words and again I can't explain why but you might have had a moment like that I knew that they were the Lord's words for me what he was saying was close your eyes let me do this you have been a control freak your whole life it's time for you to have a rest but here's the thing this idea of trusting the Lord in a real way that I'll just tell you that is not what I signed up for I thought that being Catholic was like you know the truth and you follow the rules and you have less free time on Sunday trusting God in a real way for a lifelong atheist that that just sounded crazy that just sounded irresponsible so I so I told God I was actually in the adoration chapel I went to the adoration chapel and I said I don't know how to trust you I don't even know what that looks like but I'll just tell you this here's some problems I'm having God and you can tell me what I'm supposed to do about them I don't have a house I don't have a place to live you know what that's like that I have nowhere to go I can't afford this medicine that I need to save my life and frankly even if I did have a house we used to live in a loft downtown I don't have furniture for a house so there you go I know that you know it's not supposed prayers not supposed to be like an ATM machine you know where you get exactly what you want but I didn't know what to tell the Lord other than what I needed the next day I go to the pharmacy to get my thousand dollar prescription but I realized that morning I didn't have money couldn't afford it we just barely didn't qualify for financial aid programs sometimes the people at the pharmacy if I looked pitiful enough would take mercy on me and give me some samples so I went down there and I gave this girl my name I said I'm Jennifer Fulweiler I have a prescription and I was about to say but I can't afford it do you have any samples and she just starts typing and says that'll be $30 and I said no unfortunately I've been through this a lot it's not $30 so double check and she said no I see it right here it's $30 so I got my full prescription walked out for $30 week later Joe and I were driving through the neighborhood happened to see a for sale by owner sign that had just been up had just been put up a few hours before it was the perfect house for us it is the house we still live in today we got it so cheap from this owner who just wanted to get out of there that the mortgage company made a sign in affidavit swearing that we did not have a previous relationship with the owner because it was so cheap they couldn't believe that we got it for that price furniture I told God I was like I need a coffee table a sofa a refrigerator and lawn furniture hey you want you want me to close my eyes I mean all right close my eyes this is what I need my dad got a job overseas had a bunch of furniture he wanted to get rid of asked if we needed any of it two of the items included a sofa and a coffee table my mom who had not known of my prayer said she was sick of her fridge had wanted to get rid of it could we take her refrigerator the owner just of the house who we bought the house from just before he left he said he was going to a condo where all the yard stuff was taken care of he left us his entire suite of lawn maintenance stuff as a gift every single prayer was answered and because I'm me I was like Lord was that from you I really I was like well I mean it could have been coincidence I mean you know atheist training coming here so I said Lord was that I mean I don't want to be irresponsible I mean did are you telling me that I can trust you because you know you've only answered 50 prayers in a row because I'm really not sure with this prayer on my lips saying Lord is this your sign that I am supposed to trust you I went back to that same pharmacy I went to pick up my prescription oh yet another answered prayer is we got new insurance that would fully cover my medicine for only a $50 copay so from $1,000 a month to $50 a month but the new interest the new insurance would take effect in like two weeks so I thought while I still had the old insurance that was now for whatever reason $30 I would just save 20 bucks I'll go down get my $30 refill before the $50 plan started so I go down there same pharmacy and I go to pick up my prescription and the guy rings it up and says oh that's expensive oh wow poor you that'll be a thousand bucks and I said and I mean it was okay because again the new insurance was about to start but I said no this it's $30 I know we've been through this you'll see last time it was $30 and he did some typing and and I said look look up when I was here last time I got the stuff for $30 he typed some more and he looks at me and says ma'am there is no record of that transaction now the takeaway here is not that God just gives you whatever you want under the Sun I mean because if it were I've been telling God I have my eye on the iPhone 6 you know if he wants to continue this role the takeaway is not that you give give the Lord your shopping list it's like Amazon you know he just sends everything to your doorstep for me what that answered prayer was was letting me know that I can trust him because before that moment I had thought of being Catholic as an intellectual endeavor like you might decide to become Republican or Democrat or libertarian I had examined the issues and I had come down on the side of Catholicism and that's all it was and what I learned with what happened in my life there is that Jesus Christ is a person and that the truth that I had always been looking for is not just a concept but is a person and that is what I want to leave you with here at the conclusion of my story the biggest thing that I learned through this crazy conversion it is the most important lesson in the world but it is so easy to forget and that is that the truth is not a concept but it is a person the living person of Jesus Christ who is with you and will take an active role in your life and he will help you and anything you go through and until you understand that you will never fully understand the truth thank you


So, I have just redrawn all the 14 states, these were the same which I used last time. So, because we need to keep on referring it, so I have just. . . The formula which I wrote last time, I had not explained that I think, I left some of you might have intuitively figured it out how it happens. So, maybe I can explain how that formula comes. So, I will start with that itself, that was p j tilde that is what you are computing, this was probability that a packet at switch output link in stage j will be passing out in time torque k. So, let me write down the expression and see how that can that we get derived actually. So, this was p j tilde which was there, this is the probability that a packet at switch output link in stage j passed actually forward in time interval and this is a basically a conditional probability. It is conditioned on that the packet has to be there at the output of j th link. So, there has to be a packet first and then it will be transferred on to this thing. So, remember the step one is when this packet goes out and after step one the packet has to jump actually from here to here. So, this one was basically after step one. So, after j plus one if the packet on the outgoing link has gone out that is a step one is over. After that step what is the probability that it will be in which state that will decide what is going to happen in step two here and that in turn will decide what will happen in step one here. So, if you remember the diagram step one when it happens here step two is happening at this point and step three is happening further that is what was usually the case. So, whenever the step I think I had drawn this figure that time two specific cases when I defined step one and two. So, when there are two stages. So, when this packet is going out. So, here it is happening step one the packet is jumping. So, step two is happening here for this packet is step packet is arriving at the input link. So, step three is happening here step three packet arrives in your input buffer step two packet goes from an input to output buffer step one packet goes from your output buffer to onward. Output of j minus one to output of next state whatever is your stage from your output the packet should be moving further that is your step one and once your output link is free then only from your input to output link a switchover will happen that is a step two for you and once your input link is free or available that buffer on that then some packet from the previous stage can come in that is a step three for you. So, basically when this movement will happen will be decide by after step one in which state this particular switches. So, I have to look at all those possible states where there will be a packet here at the input. Now, since actually this the way it will be is I have to look at now basically because this is a conditional probability that first thing the packet has to be there. So, I have to now sum up all the state after step one what are the states all possible states because state probabilities are absolute probabilities these are not conditional ones. So, find out all possible states and sum up all the probabilities combinations by which a packet will be here at this place. So, if you look at for example, state one there will not be any packet here going out I have to look at only those states where this packet will be there and then I have to find out whether packet can move on the other side or not. So, for example, if you look at two there is no packet at the input this cannot be counted three cannot be counted four there is a packet here the question is you might be connected to this link or you may be connected to this link. So, probability that this is in state four you have to multiply by half with only half into being in state probability being the probability of being in state four multiplied by half with that probability you will have a packet here and with remaining half multiplied by the same probability the packet will not be there at least only those situations or those probabilities where the packet is available here. Then I have to now sort out those conditions where the packet can move from this place to this place in step two when the step two will be running here. So, after step one what is the current state that itself governs what is going to happen in step two. So, after step one if you are in this state. So, after step two this packet will be going out. So, that should be there in numerator because of that look at this situation with half half into probability of being in state five you will have a packet half into the same probability you will not have the packet, but when you have the packet this packet cannot move further in step two because there is this is a state after step one. So, there will not be any movement. So, this should not appear in the numerator. So, I am looking into absolute probabilities in the denominator multiplying by the probabilities when of those states where packet is going to move out in a step two in j plus first stage because your condition is this after step one in this stage. So, what is the probability that in step one packet will move out that is what we are saying is p j tilde. So, we have to essentially do this enumeration. So, I think for clarity sake I am let me just write down this expression, but this time building it step by step is last time I did it in hurry I only did it in the end of the lecture. So, today we have time. So, I can do this and remember this is a conditional probability that is why I am doing it. Now, look at one one should not be listed in the denominator two I think should be listed or not no three what about four yes and half of it or one of it half of it. So, it is half and what I should write it is probability after step one and what is a state four this I am talking about j tilde. So, this is j. So, this should be state for j plus one. So, I am going to write j plus one here and then of course, this is in time tau k. So, I have to write k on top of it after time step k tk. Now, with five what should I write half agree. Now, go to six half of being in probability of six. Now, moving over to seven again half eight has to be full probability of being in a state eight what about nine all are full. So, I am going to write all of them. So, only these many states have to be included in the denominator. Now, let us look at the numerator when the packet will move out I am looking at p a tilde conditioned on that the packet happens. So, this states will actually give you essentially one over this. Technical gives nothing, but probability that this is actually all possible states which. So, this technically gives nothing, but probability that your packet is here one over that. Input of j plus one. Yes, right. Sir, step one in j th state After computing step one in j th state j plus first I am looking into the state of j plus first block or switch in j plus first stage after step one. See, then only I will know what was the state after this what will happen step two. So, this is situation after step one. Now, I am going to look into what can happen possibly after step two. So, if you look at step one that state one nothing is going to happen that state is not of our interest there is no packet. So, it cannot contribute to the probability that this packet will be there. See technically what happens is you will have always p conditional the packet has to be there and there is a conditional probability that packet will pass. So, absolute probability when you multiply by that when packet probability that packet will be there this conditional probability is coming from here actually. So, this is probability that packet is going to be there. The above one gives the absolute probability. So, absolute divided by this thing this is like probability that packet will pass divided by probability that packet is going to be there. So, this is absolute probability you have to divide by the probability that packet is there that will give you condition. So, this is the probability that a packet is present at the output of j th stage that is why I have written that thing denominator because what I am going to write in numerator is nothing but absolute probability of packet being going out, but I look I want to have a conditional probability instead of that. So, look at state one cannot come here because there is no packet can move out. So, it cannot contribute packet two there is no packet. So, packet cannot move out it will not be there state three no state four packet will move to the second step the output of j plus first. So, j plus first will be executing when it goes through second step this movement will happen and then the step one will be happening in j th stage. So, here step two will happen see it will be like this. So, the step one has already finished here. So, now when the step two will be executed this will be moving on this side and then that time there is step one will be running here that is what this p j tilde is about after step one packet moves out. So, after step one what was the state will decide what will happen in a step two. So, that is why I am going to write in numerator the state probabilities after step one. So, four will contribute and how many times with half probability only it will happen because I may be connected to this link or this link if you are connected to this link there is no packet going out when you connected here then only packet is going out. . . . . . .So, I am all equivalent states are being represented see no one state can be mapped on to another state by any kind of rotation flip or whatever it is that has been ensured. So, you will have half of p after one four j plus one k. Now, look at step state five even if after step one in this particular stage you are in a step state five a state two when we is going to be executed a step two is going to be executed packet cannot move because one packet will be present here. So, this cannot contribute to p j tilde. So, I should not put it in the numerator even when packet is present actually at the output port. So, five will not contribute in the numerator look at six packet will move out after step two in j plus one and packet will be emptying the outgoing buffer after step one in j th stage. So, that has to be included the six one and that will again happen with half probability because I can be connected to either of the two links input links look at seven no packet cannot move out. So, it should not be listed in the numerator state eight only one will move out and you can be connected to any one of the input links with equal probability 50 percent chance it will move out remember this one is eighth one is listed with full here it is not half in the denominator, but in numerator it will become half because only one packet can go out not both. So, I will be listing ninth both of them will go out with 100 percent thing. So, it is there is no half here in this case 10th nothing will go out. So, it should not be listed 11th only one will go out. So, it will be half not 10th it says to be 11 sorry 10th nothing can go out 12th again only one can go 13th nothing can go 14th nothing can go. So, those will not be included in the numerator. So, this gives you p j tilde. So, I think now this is more clear in the second iteration last time it may not have been and most important thing at the end the last link last stage. Since, remember now what the case which we are looking is when t select is 0 and sorry t passes t passes 0 that actually means when two packets are there in the last stage and they are both can go out. This buffer will always be empty because packet transmission takes 0 time is instantaneous even if both have to be switched out they both can be read instantaneously and taken out of the buffer. So, what will happen is this is n minus first stage or n minus n minus 2 because we have 0 to n minus 1 sorry. So, it means p j p n minus 2 is 1.0 that is a boundary condition. So, when we will iteratively do the solution this will be the boundary condition at in any stage q j will be defined as p j tilde. If p j is probability of packet going out q j is it will not go out it will remain there. Only one boundary condition that is yeah I am now coming to the other formulation also. So, all of them will be used continuously in tandem then I will give the process the method. So, I have to give all this then I have to give transition probabilities and then I have to give the method which is has to be executed. This all is done in a computer program it is not a simulation it is a computation. But it is a iterative computation it will converge at some point of time and when it converges that time you take out all the results and find out the throughput. So, that is the procedure there is no close form solution. You can write 1.0. You can write 1 not an issue probability cannot be. So, somehow we always write it as 1.0. This probability both t pass and t select are present in this probability. T select t pass is 0 t select is complete what we call t delay. Sir if t pass is 0 then 8th state those two packets can be sent outside. Depends on in which state you are in which a stage you are. If there is a last one see for example, this states cannot exist where the outgoing ports outgoing packets are there these states cannot exist in the last stage because outgoing will be immediately emptied out. This will and in n minus 2 stage this is going to be there for some time because where this is selecting these packets have to be held here. When the t select is finished then only they will be removed. So, you can actually have these all the states are possible in n minus 2. But in n minus 1 only this state is possible this state is possible and these two states are possible. No other thing will be existing. So, 1, 4, 8 and 9 that those are the only possible states for the switches in n minus 1 stage. From there we will do I will actually tell you how this backward calculation is done. Sir, we have to do this computation process. It is a iterative computation process that is the only way. Yes, it will be computation and simulation. Simulation what you will do is you will actually create you will actually generate packets. You will let the packet flow through. See it will be like kind of you will have a clock a universal clock at first clock there are no packets then the clock goes from 0 to 1. You will generate packets at the input and then what happens in that clock you will do find out the whether conflict is not there or not there you will actually move the packets. You will maintain a data structure which will maintain the complete state of the switch and now this state will be evolving on the basic principles and you will keep on moving. Ultimately at the output you will observe how many packets have passed successfully. Now you have to observe for multiple clock instants. There is nothing like what we call a time after which you find out there is a convergence. There is nothing like convergence there and in simulation there is another very tricky thing. Here I will just look for stabilization. When convergence happens I will just take the results in this computation. In simulation there are no packets so it is not a steady state. So initially there will be a transient. So I will discard that actually all that observations and after that I will observe per unit time per unit slot how many packets are successfully going out. I will keep on making observation. I will observe for certain number of time slots. It is like you implement actual switch but not in actual reality but in computer program. So you are simulating the algorithm itself not computational algorithm but operational algorithm by which they are being or they are operating. So whatever is that patrinette that essentially you will be implementing there or equivalent of that but the paper does not actually paper do have simulation results which are matched with the computational result. These are very precise. It is like I ask you what is the how you will compute the value of pi. So one very simple way how you will do it through simulation. Can anybody tell how you use simulation to generate the value of pi? We keep on drawing circles and keep on taking the ratios. No that is a different thing by simulation. Can anybody think? We have circle of radius r and then r arc length. That is fine that you know all by mathematics. You do not need to remember you only need to remember that the area of a circle is pi r square and area of a square is r square and you always know that if I make a square I can always create a something like this. So what is the area? 4 r square for the square. Circle has area of pi r square. So what is the ratio? Pi by 4. Pi by 4. Suppose circle to this thing. Write a very simple program with uniform probability I do a coin toss and I put a point. So I choose a random number generation and the points will be just randomly plotted. You put 1 million points. How many fraction of points will be there inside the circle and how many will be total points which are plotted inside this? So find out the fraction which lies inside the circle and that fraction most likely will be nothing but pi by 4 and once you know that fraction of points which are there inside the circle multiplied by 4 that will give you value of pi approximately. So this is a pi measurement through simulation. This is not computation. So I think now you understand what is the difference between simulation and a computation. Otherwise you write a series thing and do it or do whatever calculations draw circle make length or whatever it is that is a computational process. But pi there is a closed home solution no it does not exist. So closest approximation is 22 by 7 which is we all know that is approximation. But best is draw a circle measure the length find out the radius length take the ratio and that is a value of pi. Larger the circle better will be the estimate. So that is a computational procedure. So I am not still simulating you have to understand this is still a computational procedure. My students do have a tendency of mixing of the two things. So usually computation simulation is nothing but whatever happens in real life from those observations you make an estimate that real life experiment is actually simulated in a computer program is conducting experiment in computer program through a computer program and then from there making observation. So usually random number generation will be part in partial of any simulation process. I am not doing any random number generation here. So it cannot be simulation and there is no statistical estimation. So this is a computational procedure. So paper actually does give both results and they actually match pretty well. So this is seems to be a logically right way of getting the solution. This is nothing but a iterative method of getting to a solution of a Markov chain which I explained earlier in the last class. Where I give it for a Markov arrival Markovian departure and single server queue with infinite buffers. So how we can get a solution to that. So it will essentially convergence will happen and that time you get all state probabilities. So for all complicated Markov chains I think this is a very good method and here what the nice thing which has been done is how many switches are there in a stage those are immaterial. How many stages are there those what matters. So all state probabilities within a given stage are all for every switch it is going to be same. That is assumption and we have built up the relation between state probabilities of various stages. That is what I am doing actually now. So next probabilities which we need to compute is probability that a packet will I have to essentially erase this. See because remember state transitions how it is happening. When I am looking at jth stage when a packet is going out state transition happens then from there from my input link to output link there is a packet movement that is a step 2. Step 3 when the packet will come from the previous stage. So I have to look at the state transitions. So this first transition after step 1 in this thing happens as per this relation p a tilde it depends on that. So if the packet moves out or does not move out we will decide what will be the state transition matrix here. So if what was a state at t is equal to 0 or just before the first step and what happens after the first step. First step is this moving out when this has a step 2. Then once we know what happens if the step 2 is run. So if I know the state after step 1 whether packet is there at the input what is situation at my output that will decide. And once if I am available what is the chances the packet will come from the previous stage that will decide for the step 3. So this is for that situation time interval talk. So again I have to write down all the state probabilities. This will also be called pj tilde this also will be called pj. Now it is not pj tilde it is pj bar sorry. So now what is happening is these are situations. So I am now looking at the states here and step 3 is step 2 is running here. And what happens in the step 2 will decide what will happen after step 3 here. And what will happen in a step 2 will be decide by what was the state after step 1. So now let us look at it. I am trying to find out what is the chance that a packet will come from this input to here. So after step 1 if this j minus first stage in a state 1 can a packet come here. First thing I have to find out I am actually I am having an empty state here there is no packet. I have to look at only all those possible states after step 1 here. Then only I can figure out what will happen in step 2. But first you have to check whether packet is there or not. Packet should not be here. So a step 1 a state 1 no packet at the outgoing port. So it is free. So I have to list that actually. But step 1 the packet will be sent out from the output port. Which one? After step 1 for step 2 output port will be always green also because in step 1 whatever the existing packet is already sent out from the output. No that can only be sent out it is possible if you look at earlier case. Sometimes packet cannot be moved out again after step 1. If there is a packet in the next buffer you cannot just simply push you have to just keep on waiting you have to just keep on waiting. So after step 1 what are those states where this port is free? I have to look at those. That has to go in the denominator. Numerator will be those probabilities of those states where the packet will move from input to output of j minus 1. So look at this. So we will not consider anything which is not available. I mean packet has to be there there is a probability of that also in the input stage shall we not. I am only looking at the relations as of now. After step 1 what is the state here? That will decide what will happen in step 2. Because when the step 2 will be running here step 3 will be happening here. I am looking at the probability after step 3. Probability that the packet will arrive in the input buffer. This can only happen this will be decide by what happens after step 1 here. What is the state? That will govern the computation of this probability. So after step 1 if there is a state 1 j minus 1 this is important. Both of the outgoing links are free and I can connect this is the remember earlier case was like this. We are sitting here in jth stage. So you are looking at j plus 1 at that time. We are looking the backward direction. So with probability 1 I can be both of these links can be connected to me is not half both with equal probability. So both times it will be empty if after step 1 this as if after step 1 this is in state 1. So I will write this state 2 half because there is a half chance because this is after step 1 you are in state 2. So you can be here or you can be here packet can be any one of the 2. So with half probability you will actually may get it empty or may not get it empty. And if it is not empty the packet cannot come if it has to be empty first. So I will write now half of p 1 2 j minus 1 k p 3 p 3 should not be there because then my input buffer link itself is not input buffer is not free it is already occupied. So no packet can come so it cannot contribute 4 yes it is there and 4 means both. So p 1 4 there is no half in this case state number 5 half state number 6 again half state number 7 no state 8 full state 9 full state 10 is half state 11 is half state 10 will be half right 11 also will be half 12 also will be half no 13 14 cannot be half half they cannot exist we got everything right here the upper ones we have to look state 1 packet cannot come in packet cannot go actually in step 2 there is no transition possible. So this cannot contribute packet state 2 cannot happen state 3 is anyway not possible state 4 yes if the switch at j minus 1 is in state 4 after step 1 in a step 2 packet will transition. So this will contribute to p j bar so that has to be listed but this will be happening with half probability because your stay your switch can be connected to any one of the outgoing. So it will be half of p 1 4 j minus 1 k p 5 no there is no possibility if you are in that state packet will not come in packet 6 yeah state 6 this is going to happen with half probability then after 6 7 nothing 8 you will get with half probability a packet while in the denominator it must be actually full probability because both outgoing links are free 9th yeah it will be full 10 nothing will be there no I mean 11 yes packet can come it will be half 12 also will be half but both cannot go simultaneously only one of them will be there is got absolute probability then the question is you can be either connected to this or you can be connected to this with half probability no see j switch is connecting to j minus first. So you should have connected with half here you will get a packet if you are connected you would not get a packet. So that is why this is listed now with this thing this is remember what is happening is this is transition after step 1 this tells what is going to happen in step 2 essentially. So I am looking into arrival probability here so what is going to happen in step 2 I am looking at that now for this the boundary condition will be p 0 bar means the first switch the input of that I am looking at a maximum loading condition if this is free then the probability that arrival will happen is 1 that is a maximum loading condition again sorry I can write 1 and of course q j bar 1 minus p j bar that condition will be there. Sir in this zoom denominator 1 by 2 p 5 state how will be denominator in denominator. 1 this one look at the fifth one see you can be connected here then your switch was state that buffer was not empty. So that should not be counted with half probability you will be connected and you will have a buffer which is empty. If it is connected like that it will be. Conditioned on that in j th stage my switch input is free there is no packet there on that condition what is the chance that packet will arrive in step 3. Actually in state 5 sir step 1 has not happened in j minus 1 in state 5 sir step 1 has not taken place in j minus 1 stage. Does not matter I am looking at after first stage whether what is the chance of being in this state if you are in this state you are this is free but no packet will be transiting this is not there in numerator this is only in denominator. So I am looking at what will be probability that a packet will be arriving if my input is free I am estimating that. So this contributes in the denominator only 50 percent when the that input link will be free for that it is contributing to that actually it is not there in numerator. Now let us come to the state transition tables these two things essentially we will be handling try to build everything here this figure is still has to remain there. So I call this transition matrix as p j in the j th stage this is after step 1 going from initial stage is m and you want to go to initial state is m and next state is i after step 1. So I make the probability is going to 1 2 3 these are the states only 14 are required. So if you are in state 1 what will happen after step 1 there is no packet at all outgoing thing. So there is no question of it is going to be certain probability p j. So with probability 1 you will remain in state 1 only. So we will keep this as 1 so I am going to put only the values which are existing rest everything is 0. I am not filling up all the cells here all other cells will contain 0 except the way where I have written the values. Now look into state 2 state 2 will remain now the probability that packet which is sitting there goes out is p j tilde. So if that packet goes out you can come back to state 1 if that packet remains there you remain in the same state. So it actually means this will be p j tilde no this should be a p j tilde you will come from 2 to 1 you will remain in 2 with q j tilde rest everything is 0 not clear. So you have a switch and if it is in state 1 this is a state 1 and what is step 1 packet from at your outgoing port goes out and if you have a packet here this will go out with probability p j tilde and it will remain there with probability q j tilde which is 1 minus of p j tilde in general stage because if it is a n minus second stage packet will certainly go out there is not an issue after step 1. And if there is another one so this will be going independently with p j tilde I am going to talk about 1 packet condition if the packet is there if there are 2 packets both of them will go out what is the probability p j tilde square p j tilde square only one of them goes and one of them remains multiplied by 2 there are 2 possible combinations both of them remains there p j tilde square and all these 3 events will lead to certain state transformation. So there is no packet there is no packet here with certainty with 100 percent probability or 1 probability it will remain in state 1 only that is what I have written there. So this is your m and this is your m to i so m to i transition this is initial state this is next state after step 1 in jth stage. So this is a transition probability in jth stage after step 1 for going from mth state to ith state. So you can actually write going from state m to state i in jth stage after step 1 and this is nothing but the probability and this is what is this transition you build up the matrix you find out from each state to other state what is a transition probability. So 1 to 1 is obvious actually if you are in state 2 you will have only one packet sitting here if this packet goes out what is the probability of going out p j tilde. So if it goes out in which state you will be state 1. So I have written here p j tilde from 2 to 1 this is the probability that you will be transition and 1 minus of that packet will not go out you will remain in the same state if you remain in the same state that is what I have mentioned here and remember rest all these blanks will be 0 you cannot transition from state 2 to state 4 for example that is not possible look at a state 3 now which contains 2. So with probability as all of you have told p j tilde square all both the packets will go out you will come to step state 1. So you write here p j tilde square n f of course only one packet goes out and one remains you will remain in state 2. So that is 2 p j tilde and q j tilde of course this is very small. So maybe I can write it each event is happening independently. There are 4 possible cases there are 2 possibilities here and there are 2 possibilities here with equal probability. So one case is p j happens other says p j happens here q j happens here p j happens here p j happens here q j happens and here q j happens here q j happens each possibility is 1 by 2 1 by 2. So these 2 are same they always lead to the same state so that is why 2. Actually this is a tabular description sir of Markov chain tabular description. It is a transition probabilities is still Markov chain has 2 things the state probabilities and then transition probabilities Markov chain has these boxes. So there is a probability of being here and this is the transition probabilities. But you can do the same thing with. That I am not doing tabular sir this condition that does. No because I am not actually making any Markov chain. Yeah technically yes I am doing the same thing you are right. Technically I am doing the same thing. We will also have a transition matrix for that bar state p j bar similarly which you just derived now. A transition matrix. Yes sir. It will come this after step 1 I have to draw t j 2 also t j 3 also there are 3 matrices which will be there they have to be maintained as data structures during your computation. Simulation is far simpler actually remember if you want to write a program for simulation that is going to be simpler than this. But the good thing is that you do not have to remember all the possible switches which can exist in a single stage. So only states you have to take care and you have to take care of stages. State to state actually transition will be there and of course q j you will both of them will be remaining there so 3 2 3 q j tilde square. So 4th will be similarly if you are in 4th one then what will happen? Step 2 may happen. Step 2 may happen. After step 1. Step 1 is packet from your outgoing port goes out. There is no packet at outgoing port. So nothing will happen after step 1 only step 2 can make a transition in this. Step 1 cannot make any transition in this state. So this will remain in the same state this cannot do anything. So 4 will remain in 4 with probability 1 5 with probability p j this can be converted to 4 with q j it will remain there that is it. So with 5 can be converted to 4 p j thing and it remains 5 with q j tilde. This is sparse matrix actually. 6th same way after step 1 packet can go out or packet can go out. So if the packet goes out this becomes nothing but 4 otherwise it remains 6. So it is if it goes out it will remain p j tilde 6 it will remain q j tilde. Of course 7th state this can be converted to 4 p j square 7th if it is there p j tilde square and if one of them goes either 5 or 6. 5 or 6 with equal probability. So you have to write p j q j tilde p j q j tilde this has been remember the complete row when you sum up has to be equal to 1 sum across all the rows has to be equal to 1. And of course then q j if you have it will remain 7 8 what will happen 8 will remain 8 not an issue 9 will remain 9 10 will become either 8 or 10 will remain 10. So 10 will remain 10 will become 8 with probability p j tilde and 10 will remain 10 with q j tilde 11 11 will either remain 11 or will become 9. It will be 8 10 row 10 row 10 row it should be 8 column p j tilde. P also has to go to 8 column. Same you can do with 11 11 goes to either 9 or 11 either become 9 or remain same 12 will either remain 12 or will become 8 13 there will be 4 entries for this. So either it will become 8 with p j square and p j q j either it will be 12 or 10 10 and 12 p j q j and of course 13 none of them goes 13 cannot go to 9 same directions the same 4. See direction you have to always take care 13 cannot go to 9 13 only can go to 8 can go to 10 or can go to 12 or remain to 13 14 14 can go to 9 11 14 can go to 9 11 may go to 9 where is 9 yes 11 it is 2 p j q j tilde that is a complete transition matrix. Similarly, you can draw the transition matrix for after step 2 now important thing if you have this so after 0 step you know the probabilities this basically is the beginning of the slot. You can multiply it by now transition probabilities after t 1 j m i sum over all probabilities I think now you can appreciate if I know the state probabilities in state j after 0 I can compute from there my state probabilities after step 1. And these are the transition probabilities which will come and transition probabilities now depending on so this here where I am actually finishing next we will look into the next state transition diagram which is after step 2 and then step 3 and so on and then actually the method of calculation. So, that will you will covering in the tomorrow lecture morning.


Bonjour tout le monde, hello everyone. We were a country that stands united in our effort to defeat a pandemic, to save lives, and to help each other make it to a better day. But yesterday we were jolted from that common cause by the senseless violence and tragedy in Nova Scotia. A gunman claimed the lives of at least 18 people. Among them a woman in uniform, whose job it is to protect lives even if it endangers her own, Constable Heidi Stevenson of the RCMP. Constable Stevenson died protecting others. She was answering the call of duty, something she had done every day when she went to work for 23 years. This happened in small towns. Porta Pic, Truro, Milford and Enfield are places where people have deep roots, places where people know their neighbors and look out for one another. There, everyone knows at Mountie because they're everything from police officers to social workers to teen counselors. Now these communities are in mourning and Canada is in mourning with them. To the grandparent who lost a child, to the children who lost a parent, to the neighbor who lost a friend, we are so sorry for your loss. Such a tragedy should have never occurred. Violence of any kind has no place in Canada. We stand with you and we grieve with you and you can count on our government's full support during this incredibly painful time. Yesterday, at least 18 people lost their lives after a shooting on Newcastle Avenue, crying families and their loved ones in tears. To the grandparent who lost a child, to the children who lost a parent, to the neighbor who lost a friend, to the neighbors who lost a friend, to the neighbors who lost a friend, to the neighbors who lost a friend, to the neighbors who lost a child. Such a tragedy should have never occurred. Violence of any kind has no place in Canada. We share our debt and we feel sorry for you when you live extremely painful moments. I also want to wish at full and speedy recovery to all those injured including another RCMP officer. is a vast and sweeping country filled with long stretches of lonely roads. With unwavering courage and compassion, our Royal Canadian Mounted Police patrol these roads to keep us safe as they have for over a hundred years. I know that from coast to coast to coast the women and men who wear the red surge in service to Canada are grieving deeply the loss of one of their own and one of our best. This tragedy is a painful reminder of the risks all of our first responders take to keep us safe. Of the sacrifices they make every single day to protect our communities. Paramedics, doctors, nurses, firefighters and police officers. They're always here for us. They've been stepping up through the pandemic and yesterday in Nova Scotia they showed that bravery. I want to take a moment to thank them all for the professionalism and their courage. Many of you are already working overtime because of this pandemic. Our communities need you now more than ever and I know that can weigh on you at times. These are exceptional circumstances yet you did what you always do. You ran towards danger without pause, without hesitation. You put your life on the line on behalf of all Canadians. Thank you for your service. I want to take a moment to thank all the first responders for their courage and their professionalism. You're already working hard because of the pandemic. More than ever, we need you and at times it must be a responsibility that is hard to bear. Despite the exceptional circumstances, you did what you used to do. You didn't hesitate to risk your life to save others. On behalf of all Canadians, thank you for everything you do. This day is made all the more difficult because of the precious lives lost and the senseless act of one person. Just how could this happen? We may never know why, but we do know this. No one man's action can build a wall between us and a better day, no matter how evil, how thoughtless, or how destructive. Canadians are kind and generous. We are there for each other and we look out for one another. As families grieve the loss of a loved one, all Canadians are standing with them. The pandemic will prevent us from mourning together in person, but a vigil will be held virtually to celebrate the lives of the victims at 7pm on Friday through the Facebook group Colchester supporting our community. As we learn more about what happened yesterday, it's important that we come together to support communities. In the hours since the tragedy I've spoken with Premier McNeil, RCMP Commissioner Lucky, Minister Blair, and many Nova Scotians including our Nova Scotia caucus, we will continue to work together to see this through. I know that people have a lot of questions. This is an ongoing investigation, but I can assure you that the RCMP and local authorities will keep you updated. I want to ask the media to avoid mentioning the name and showing the picture of the person involved. Do not give him the gift of infamy. Let us instead focus all our attention and attention on the lies we lost and the families and friends who grieve. Although the pandemic will prevent us from mourning together in person, a virtual vigil will take place online at 7pm on the Facebook group Colchester supporting our community. It will be an opportunity for us to celebrate the lives of the victims and to be there for their families and their loved ones. I hope that many of you will join us. Thank you. Thank you. I want to close today by addressing all the kids in Nova Scotia and right across the country.  You'll see it in your neighbors and in Canadians in the days and weeks and months ahead. This is a difficult time and it can be a scary time too, but we're here for you and we're going to get through this together. I promise.