Hello everybody! Hi! Thank you here! We're so excited that you're here for... Treasure Island! Wait, what play was that? Treasure Island! Treasure Island! Treasure Island! Has anyone here read the classic Treasure Island by Robert Louis Stevenson? Yes, you're gonna see something kind of like the book you read. But I've changed it quite a bit. And my name's Reba, and I'm the director of this play. And I also adopted it, and I had a lot of fun with the story. It sounds like actors are still backstage doing their warm-up. Can you hear them? Yeah? They're counting! It's so true! Everybody, enjoy the show! Thank you so much! The bed bow in. Welcome, sir. I run the place with my boy Jim over here. It's just the two of us. You can tell me what you need. Are you looking for a hot meal or a place to stay? Good-bye, man. May I ask you how you heard about the place? I shipped rock to Ny-Astra on town. People said your inn was mighty fine. That'd be fine. It would offer me a peaceful place to clear my head. I haven't had a hot meal, and I don't know how long. Or a place to rest my head that wasn't wet and moving side to side. How long have you been up to sea sailor? Months. Maybe years. What boat do you sail on? There's been more than a few, my lady. Listen, this is a fine establishment. I won't have any trouble. No shenanigans, no wrapped scallions, and especially no pirates. Do you hear me? Oh, not I, man. No pirates. I can't have the trouble.  I'm tired. Oh, so welcome, sir. Come in. Oh, nothing.  Of course you leave treasure, don't you? Gold bars, diamonds, rubies, stuff that make you rich. Never have to work a day in your life. The ocean, that blue-ish greenish mass of water. You've been to the beach before, right? Have on a ship, on a clear day. See the horizon, stretch beyond your dreams. And you can't quite tell where the sky stops and the water begins because they're both kind of same color. Then they're in a daze when the ocean roars and buckles and waves hide. Your math is more beautiful than any ruby or sapphire you'd ever see. The ocean's a little full, you know. Felt as though it's a call. A call to adventure. Do you smell it? Jim, what are you doing dreaming again with all these people staying at the inn? You said you'd help me. You gave me your word. What do you need? Check on the captain, room six. What did you think? That I was a pirate? No, I just yearn for adventure. I look off to see and imagine my mom calls it dreaming. I would love to be on a boat sailing the high seas, but I have work at the end. Your logins. You a cabin boy? No, I work at the end.  A young wee thing. I'm not small. Puppy. I'm really tight-brown, sir. I look after this inn with my mother. You protect your mother? I do, sir. I wasn't fighting you after all. I do, sir. Can I get you anything, sir? Don't call me, sir. It doesn't suit me. Call me captain. Captain? Captain Billy Bones. Yes, Captain Bones. Fill my cup.  You keep on the lookout for a seafaring man with one leg. I'm looking for a one-legged sailor. A one-legged? He's got a bird on his shoulder and one leg made of wood. You ever see that man? You tell me. Why? I can run the other way. That's why. Take my telescope, boy. Are you? He has the best adventure stories. You let sword fights. He's had them. I was saying walk the plank. He's said it. Mutiny. He's never done that. He's told me all kinds of things I need to know about sailing on ship. So I'll know when the adventure calls. Come, sir. Are you looking for a hot meal or a place to stay? I'm looking for someone. What? Who? We don't want any trouble here. Just a hot meal now. Right away, sir. Jim, a hot meal. Come here, sonny. I want to go closer. This is your table from the Nate Bill. Well, Nate Bill now lets his heart. You know me? After you now for three days, I finally found you, Billy. Billy Bones. You made off with Captain Flint's map. You tell me where that map is. I don't know what you're talking about. Really, I don't. Don't play with me. You took Captain Flint's map and you're after the treasure. Did you really think you'd get away with it? Get away with what? Easy now, old brother. Just wait until after sunset. We'll be back for you. Jim, I'm going to get away from here quick before I slip the black spot. You see chest thereafter. Take that chest and run. You hear me? Yes, sir. Close to mine. You do it, Jim. Take the chest. Jim Hawkins. This is Hawkins. Those cut- What do you say, Hawkins? Bristol. What's his name? Captain. And since you found it, you should be the one. Oh, what a true honor to hold the map, Scarce Lottie. I promise to take very good care of it. Silver's office. I just got one. The other's in the laundry. Make home for your ship. You ought to be certain. You being the captain, aren't you? Captain Billy Telescope, looking for one-legged sailor. Could this be him? Can I help you, sir? I was just asking your captain if I could come upon your ship. I'm not the captain, sir. I am the cabin boy.  I don't think you have any business on my ship. Jim, what did this man do to you to warrant such rudeness? Forgive me, gentlemen. I am just an old sailor. Hobbled on to get a smell of the salt air. What a ship this is. Do you mind if I come aboard for just a spell? It's been a long time since I've been on a ship so grand. This way, man. Silver's the name. Long John Silver, they call me. Are there many ones like a secret man? A country school of them. Just like birds in the sky. I am sorry, Mr. Silver. I am so sorry for my fondness. No hard feelings, chap. You have a boat to protect. As you can see, Mr. Silver, here we have the captain's quarters and right next to it, the galley. What the galley? Why, it's the prettiest kitchen you've ever seen. What a lucky cook will work in there. You don't have a cook.  Cooks all over England should be clamoring to show you their skills. But do you have experience in this job, Mr. Silver? Mr. Silver, I do. I've cooked on many a boats, but never a galley. So fine. You should hire Mr. Silver. Oh, I wouldn't. The job is yours if you want it. I won't disappoint. Cabin boy. What's your favorite food? Cake. Then we'll have cake every night. I'm in need of a whole crew. I hire folks, so they don't seem to come back. Oh, no. Well, ain't that a shame, Mr. Squire. How many nights you be needing? Four to six, but they need to be honest sailors. Everything on this voyage needs to be above board. I? No buccaneer types. You'll be scared of pirates, eh? I may know some. I'll spread the word. Gather them up. You know this one, that? No, but I can learn. A gentleman. The Hispaniola. That's the ship. Our ship. Aye, sunny. The young cabin boy hired me to be the cook. We're having birthday cake every night. Let me introduce you to my friend's gym. This is William O'Brien. Lost his ear defending a kid. The kid got the best of him. How do you do? And over here is my friend, his really hands. Sailor Captain Flint, the black bearded Edward T. Many greats. This, of course, is Patrick O'Hare. Quite a musician. Play something for us, will you Pat? What were you giving me, Longdraw? They weren't being any other hands. On this boat, George. Would they? Longdraw. Only good. Honest. Inquired boat with a squire named Trilani. Mental folk in this establishment. You hear me? You're not welcome here. Wait here, baby. I don't upstairs to get my bag. I can get my bird. Your bird? That was an adventure. I do hope you can... Join us? Where are you going? You gotta know your destination before you set sail. Oh, we do. We have a map. Gym. Meet Captain Flint. A remarkable bird. I named her after my old captain. Your old captain was Captain Flint. Now this bird, Jim, is two. Maybe three hundred years old. They live forever. And no one has seen more wickedness than this bird. She sailed with Captain England. A pirate. Spent a Madagascar. Malabar in Portobello. She was in the fishing of the red plate ships all because of flora. That's where she learned how to survey. And no wonder, Jim, gave fish to 350,000 silver pieces. What would you do with all that silver? She was in the Bay of Bengal, and with Captain England himself, aboard the Portobello ship carrying diamonds. I was looking at her. I think she was a baby. He says that she's seen a lot of treasure. We're a little break and sniff out treasure, Jim. Even if you're not looking for it. She's like a compass for gold. Good to have when you go on an adventure like ours. But we're not looking for treasure, are we? Of course not, boy. But if we were to happen upon it, it wouldn't be a bad thing. Now what? I suppose not, sir. Because she bites her. Narrow, you nipple. Oh, the squire that we saw, old black dog. Why wouldn't I? You don't want to worry a man like Squire Trillani unnecessarily, Jim. He's got a lot on his mind right now. A lot on his mind. My lips are sealed. Maybe.  Black dog, black dog. You flambrain. Take this idea, Margaret. There's a superstition that says it's bad luck. I don't care about your superstitions. I will work as hard as any man aboard and more so. Only you will know my secret. Please, let me be close to my boy. You're so great as the crashing of bricks cause the sea cheese a call. You're standing at the seafaring light cause the sea cheese a call in my name. A good crew, mate. He's got a mustache. Black dog doesn't have a mustache. Call me white twirl mate. Nice to meet your acquaintance. Oh, all right. Not much of a seafaring move, but he's good at being a doctor. What does a doctor do? Well, I cut people up and put them back together. Cutting people up is not fun. To the squire with my confluence, gentlemen. Attention, two gentlemen. Mass knows more about this voyage than I do, and I want the truth right now. What are you talking about, Captain? I was told this was merely a trade ship to import spices from the south. A search for buried treasure. Who said? They know we're on a voyage to find Silver's bird has been squawking about it. Was it you, Jim? This is a very risky venture. You as your captain on two conditions. The first condition is that the map is locked up in the safe box, and I have the only key. Jim, it's in the best interest of all. Jim, small it, that's map. Second condition is that we cut the tips off their knives and enlist you men to help me. I orders. Why are you men standing here? How will I cut the meat, sir? I must have sharp knives to prepare the food and to change buttons with you after we set sail. Honest man aboard, it's Captain Small It. Honest, sure. Put it down right, humble. Captain's nicer active better than he could. Look down there, boy. What do you see? A shark! That's old Nicodemus, the shark. Why is he following us? We've been feeding him ever since we left the port. We have? With potato peelings. A voyage from, John. Would you like to come live with me? See, I'll have money. Just remember me. Anyhow, those can visit, right? Will all these be mates? Certainly we will. No, it's fit into the sea stuff. It's always lucky to fit into the ocean when you're on a ship. Just as long as we don't fit on old Mr. Nicodemus because I don't think he'd like that very much. He might develop a taste for you. Just joking, lad. Old Mr. Nicodemus isn't false anymore. Is that another shark? No, that's a dolphin, a sea man's friend. Dolphin over the bow is a good sign. I guess nobody knows where we're headed to except the dolphin. Captain's log. We've been traveling for 13 days and still no land of sight. The crew is abated now. Aren't men a fortune as I originally suspected? On the ship forever. Are we lost? No, Jimmy. Captain Smallnose is going. And he's keeping us right on track. How do you know? Look up there. What do you see?  I see a soup ladle. We're in a nice clam chowder. Do you see it? Do you think it's a sky? No. Just connect the stars. See that big, different spoon? Oh yes, I kind of do. I do. All right, there. The flow of that nice sky chowder goes straight to the north star. There's a W. No, you've gone too far and spilled soup on the Queen Cassiopeia. That's the north star right between the Queen and the soup. But that's behind us. And that tells we're not headed north. Are we? Look that way, between the bow and the starboard side. There are three stars. That's the belt of Orion.  Follow the sword. And that, my dear friend, is south. So we're headed south. South-east. South-east. We're the treasures. I mean, south-east is where we must be going. You can smell the treasure. I can taste it. What does it taste like? A future. Man, my own island away from the law. To the future! This gentleman walks the plank. Leave this, sir. We've got to get our hands on that map. Be alone. One of us. A man of fortune. Yes, sir.  Old Firebeard himself. Firebeard. I've never heard of Firebeard. Sure you've heard of Firebeard. You've never heard of Firebeard. He called him Firebeard because his beard is, you know, red and once he set fire to it, I imagine you're looking for a share of the treasure. Sure he has a long job. All for one and one for all. He deserves it, just like a lot of us. Very well. Plenty enough to go around, I say. We strike tonight. No. No. Not tonight. Small is a good navigator. We've got to use the skills. When do we strike? We strike on the way home. Sending them all over the board in one fell swoop. We lost them in a storm when we sank. Limit the struggle, cut our losses, and go home with the treasure. Not as comfortable as a Firebeard. Look at the job that I say. Mom's treasure's mine. And mine. And mine.  Mommy! We send the crew ashore alone.  It's our only option. Doctor, talk some sense into the boy. His name. It's all this money, but it's our own. Tonight there's nothing that can be done except to sleep. Tomorrow we use the map to bait them to the shore, and then we head home. Good night, gentlemen. And return to the boat before the captain wakes up. He lets the key to be on purpose. Slip into a dingy, and we're ashore. We fast before the break of day. We'll harm you. You and harm old Ben Gunn. We'll proceed in five years. My heart is sore for conversation. Conversation, fat. And cheese. Cheese. If you come back to the boat, I'll give you some. The squire's doctor. She's with her dinner every night. Squire's in doctor's, says you. All I have for dinner is goats and berries.  So I'm a gentleman ship. Yes. At least with now, we sail for the King of England. How did you end up here? I was on captain's ship. We buried the treasure, and we made a map. That's the very map we made. Then we sailed away. Maybe he'll come back for it. I was. That ship got taken by another ship, flying pirate collars. We circled back to dig up the treasure, but we couldn't find the treasure because we didn't have the map. So what did you do? We walked around in circles until one day all the hands on a board. They said, Ben Gunn, you buried that treasure. You can find it on your own. And they left me here for round. We thought a gentleman would do what said you. You got a squire on that ship? Oh, yes. A squire, and a doctor, and a captain, and a crew. Who tells the Ben Gunn war with any job? Any job at all for a rind of cheese? All the treasure I will pay for cheese and gold? Come up to my cave. Right up there. I have pearls as big as Oscar Jags. My word. I don't know what you're talking about by a Wilson squire and doctor. Everything's set. Or except that one, Matthew. Very good. But who has the map? Jim's mother. This is a woman. M'lady. Who is this ship? Who is that telling me? Have you gone to shore without the map? You can't leave the... Margaret? Take my sword. You may need it. We'll meet back here before sundown. What about me, captain? Well, you already with a weapon. Just in case.  M'lady. The squire will protect you. Aye aye, captain.  No treasure's so great as the crashing waves cause the sea, she's a call in my name. Knee of all people. I only got one leg, see. I will only slow you down. I've noticed your keen sense of direction, Long John. We need a good navigator on shore. I'm pointing to you, Jim. And the map is dry. Jim, you were worried about you. I got you, boy, nice. Give me the map and we'll be on our way.  Now just do as you're told, boy, and I won't hurt you. Just walk forward. Just follow the map. See, Jim and I, we're gonna find that treasure. It's gonna be ours. You, captain, doctor, you're gonna walk the other way now. Now you hear me. You are not to touch a hair on that boy's head. You can have your treasure, but I want him returned to us safe once you find it. What's safe? I'll manage walking this. That's just so great as the crashing of life. Cause the sea, she's a calling. Seat fair enough, cause the sea, she's a calling my name. Pirate ship now. We're waiting for Long John to come back with our treasure. Don't worry, I'll get you out of this. How? I don't see any way. Where's your sword? I left it below this. Where exactly? Under my belt. Sure. What shall we do with this land lover first? Cut his fingernail. Down to David Jones Walker. Important man. A rich man. He could have you sent to the stocks when the ship docks. All the more reason to look the way. True. True, but it would be unlucky to send him to the depths now or anchored in shallow water. Don't we want to wait until we set sail? We don't care. Do it now. That's right. This crew doesn't believe in luck, do you? Captain Firebeard. He was another story. He would spit three times off the deck. Anybody said goodbye. He would never force a man to walk the plank of land in sight. You? It's hello? Goodbye is all over the place. We have bananas on board. We let the port with them. Captain Firebeard would never stand for that. We set sail on a Friday. And this morning, this time came up red. Shiver me timbers. I like the sound of this. And of all unlucky things to have a woman on deck, it's quite progressive, but so unlucky. When I heard there was a woman on board, I thought that she would sit right in the middle of the ocean and we'd all be cooking the fishes. Who's the woman? Is it you? There's no woman here. It could be any one of us. It could all be sisters, mothers, daughters. It could be you, Israel. Or you, William. You're captain now. Untie this man immediately. Lower that power flag out once and replace it with a flag of England. Other pirates will be looking here. To up on the Hispaniola to leave. What are we waiting for? An old cook with a crutch to come back with a few gold pieces? You. Follow my orders now. Aye aye, captain. We are britches. Not without a share for myself. You don't deserve it, any piece of it. As soon as we find a port, we're putting you and your crew in jail for mutiny. No, captain. I report back to duty. I'll be your chef again, aye? No. We have a new chef. His name is Ben Dunn. And he will be serving us all the cheese we want. Over in the brig and the other pirates, too. No, you. They all maroon themselves when they found out they couldn't stand their lust to have a lady captain. Mother, is that you? Jim! What an adventure we've had! Who picked up a new crew in another port? Not Jim. I knew Ben had the treasure. I tricked you. And I kidnapped you, so I guess we're even. But it's not fair that you stand trial for mutiny. You're the best friend I've ever had. The ship is docked. The sail is short, except Ben Dunn. He's keeping watch, but you can't see it. I can't. I can't. It's a real shame, Maddie. It's a shame to be locked up like this, away from the food or the freedom. You have the key there. I do. I'm sorry. I truly am. I never meant to hurt anyone. Not anyone at all. Be a friend now, Jim. I'm off that door. After the day comes that I can help you, I will. All you need to do is say my name, and Old Long John Silver will come, hobbling aboard to help out his good friend, Jimmy. You won't be fired anymore, will you? You won't steal? From now on, my course will be an honest one. Oh, this is why. Shiver me timbers. I'm glad that happened to me. I took that before I promised not to steal. Here, you take it. No. You may need it. You have to buy food. All right. Then here, take Captain Flint. Leave her good and put her below deck when there's any women around for you. You may be needing a mate or a captain. Who'd you think would come hobbling along with Old Long John Silver? We will see each other again. Won't we, Mr. Silver? Certainly we will, Maddie. Certainly we will. Well, make along through the short. We made a good cruise home. All of us had an ample share of the treasure. We used wisely. We were foolishly according to our natures. Captain Smollett is now retired, you see? He's done got a thousand pounds, which he spends more often than we do. To that crystalline. The worst dreams I ever have when I hear the surf boong arounds coasts. Or is there upright in my bed with a sharp voice? Captain Flint ringing in my ears. He's in the ring!


Okay, so you want to do one more graph conversion. Okay, you see this. Suppose this is VT graph we have, converting to TV and then PT. Try this. A to B, B to C and C to A. Done? Okay, okay, I'll see this. Assume this line is passing through origin, right? This assumption we have. Okay, understood. So B to C and A to B. A to B, the process is what? Quickly, I'll write down this. A to B, the process is isothermal. If you have any doubt, you can tell me. B to C, we have constant pressure process, isobaric. And C to A, we have constant volume, isochoric process we have. Okay? First of all, we'll see isothermal. So isothermal, I am taking like this. Graph we have, so AB will go like this. Which one is A? Which one is B? We'll see. AB will be like this. AB is this AC, CA is constant volume, right? So we'll have C over here, go like this. C, A, B, is this correct? CA will have constant volume. AB will have volume decreases. Actually, AB volume decreases, no? See, AB will have volume decreases. Here the volume is increasing. A to B, once again. C to A will have constant volume. Constant pressure we have B to C. B to C, if it is up, then A should be down. Achha, if it's, take this as C. This is A, this is B. A to B will go like this. Then we'll have B to C like this. Constant pressure C to A like this. This is the graph we have. It's correct? C to A, you see constant volume. A to B will have constant temperature. B to C will have constant pressure. This is the graph we have. B to T, P and T. AB is isothermal, so we must have one line like this. Okay. A to B, we have volume decreases. Means pressure should increase, right? So I think I would take AB like this. Okay. Then volume is constant. So volume is constant we'll have for C to A. C to A, the volume is constant, correct? So C to A, the volume is constant. So if I just roughly, I'm drawing one second. C to A, if it goes like this, then C to B will have constant pressure. Okay. So the graph will go like this. I'll shift this graph on the left side. So we'll have this line. This is the graph we have. So we'll have A to B, A to B volume decreases. Constant temperature. Constant temperature, volume decreases, pressure should increase, correct? So we'll have this one. A to B, and this is C. AB, BC, C. Look at this graph. Is this correct? Tell me. All the process, everything you can understand here, it is coming out to be right. Yeah. So this is the graph we have, right? Fine. So this is the answer we'll have, right? You check your other things, Suresh, okay? This should be the graph we have. Okay. So you have to, you know, you have to, again, think about, you know, one process you just take, place it here in the axis, and then you see whether you have the other points, whether we should have the other points on the left, on the right, have a guess on it. You will have the final graph you will get after some time. You will have better understanding of it once you do some more practice, you will understand this. Anyways, so next we are going to start very two important process here, that is reversible and irreversible process, okay? Heading right down all of you. First, we are going to discuss reversible process. What is reversible process? You first try to understand, okay? I'll just give you the important points here, but first let's understand this. To understand this, we are taking a piston-cylinder system, okay? Copy down the graph, this figure with me, all of you. So I'm taking a piston-cylinder system here. This is the piston and the cylinder we have here. Same piston-cylinder system we have here. Same thing, there's two different states we have, that is it. So what happens here, you see, we have some gaseous molecules present here in this cylinder, and here the piston is static, right? This is static. Here we have gaseous molecules present. It's the same system at two different states, that is it. So what happens here initially, we have the external pressure, P external, and we have the internal pressure, that is the pressure of gas. Now what we do, it is here in the equilibrium, right? Now if you place a stone on the top of it, what happens then, could you tell me? If you place a stone, you will disturb the equilibrium of this pressure and this presence, yes or no? The piston will come down a bit, correct? Yes, piston will come down, okay? So you are increasing the external pressure, try to understand it this way. You are increasing the external pressure from two delta P, means P plus delta P we have. Slightly the pressure increases, the piston will come down, even if you are not observing it, it is not visible, because there is slight increase in pressure. So the piston will come down a bit, you won't even observe it, that the piston is coming down or has come down, but it is there, correct? Again, you do this once, external pressure increases, there will be a bit of compression, and then again the equilibrium went in, correct? Yes, all of you, tell me, yes, correct? So this is one step, first step, next. So this is state A we have for example, state A we have for example, and this is state B, two different state, like I said, we are going to understand. We want to go from A to B, that's what we are discussing. We want to go from A to B, we increase the pressure, what is this I didn't get you, Anurag? What is Dp by Dg is equals to zero, what is G there, what is P external? No, no, why this expression for I'm not getting you? I said, I said, we increase the external pressure, compression takes place, piston will come down, and then the equilibrium will be maintained again, the piston is balanced. Any doubt in this? Correct, okay. We want to go from A to B, again we'll put one more stone, again it'll come down, equilibrium maintained, step number two, step number three, step number four, and suppose this, you have increased the pressure to a certain extent, and then we have reached this state B. So this compression is done slowly, isn't it? One by one, one after the other, one step, then second step, third step, fourth step, like this it is going, means slowly you are increasing the pressure, right? So this process, when you are increasing the pressure slowly, we are taking an example of compression, same thing is possible with expansion also, just you go back, you remove this stone, this stone will come up, again come up, come up, come up, and you will state, you will have the state A again. So this process, whether it is compression or expansion, this way we call it as reversible process, correct? So reversible process is extremely slow process, one by one it happens. There are infinite number of steps. Here we have talked about four different steps just to make you understand, but such infinite number of steps possible because we are increasing the pressure by delta value, P plus delta B. So increase is very small, almost negligible, you won't even feel that the piston is going down, but yes, slowly it is going down, right? So this process is reversible process, any doubt in this? Yes, tell me, any doubt in this? No, tell me, external pressure is constant or variable here in this process? External pressure is constant or variable? None of you there? No, it is not constant either, you see. You have done this in physics also? Reversible, irreversible? Have you done this thing in physics? See what happens, we are increasing the pressure slowly, no, then only the compression is taking place. So one by one you are putting in this pressure, the stone you are putting in and pressure you are increasing gradually, right? And hence it is coming down. And hence what we can say, the external pressure is not constant, very important one it is. The external is not constant, it is continuously changing. Understood? Right, and now I can see problem, it was not there. Anyways, so this is not constant, very important information which we'll use in derivation also, right? We'll see that derivation, how do we do and how do we use this information? But this process, reversible process, you can have many properties of it that you should know. We have taken the example of compression here, same thing we can happen other way also that will be the case of expansion. Suppose you have state B, right? You remove this stone one by one, you will reach out to this point that is state A, that will be the case of expansion, okay? So reversible process can be compression, can be expansion, anything. If external pressure is increasing, it is a case of compression, external pressure is decreasing, it is a case of expansion, both ways possible. Some properties of this you write down, very important properties, write down, I'm sorry. Write down, it is bi-directional, it is bi-directional, means both way it can move, right? You remove the stone, you'll get this, right? It is bi-directional, this process can be reversed, process can be reversed along the same path, along the same path, third one. No external work is required, work is required to restore the system, system to its original position. Just you need to remove this, nothing much you need to do. Third property, write down, third property, write down. There are infinite number of steps and each steps are in equilibrium. Number of steps, each step is in equilibrium. It is an imaginary process, ideal process, like we have ideal gas. This process is also imaginary process, is not practically possible. Extremely slow, extremely slow process we have. Each step is very slow, most important property, P external is not constant, gradually we are increasing the external pressure. If you decrease the external pressure, that would be the case of expansion. And we can apply PV is equals to NRT at each step. These are the properties of reversible process we have. See, we are increasing delta P external pressure by a very small value. Suppose the value is P external is given, suppose 10 atmospheric P external. Now you have increased, you have decreased the external pressure to 9.99 atmospheric. You have decreased it. So that change in pressure is very small. So the process that the piston that comes down, or you have increased it, for example, I'm taking, 10.0001 atmospheric. So the piston that comes down, that shift is very small, because the pressure that you have increased is very small, large increase is not there. So it will slowly goes down. The piston will go down slowly. And that process, the movement of the piston that we have, it is extremely small because the difference in pressure is not that high. Pressure difference is what? 0.001. So pressure difference is not that high. Process is extremely slow. It will go down, attains equilibrium. Further you increase pressure, again go down, attains equilibrium. Like this, it happens in a stepwise manner, correct? And each step is in equilibrium plus extremely slow step. Any doubt in this? Anyone in doubt? No. With this graph, you will understand it in a better way. You see this graph, first of all, I'll show you. It is the reversible process graph. We have pressure volume diagram, for example this. So when external pressure increases, we'll have compression or, you know, quasi-static is a bit different from this. Quasi-static is a kind of, you know, a reversible process only very slow, but there we'll have friction also. Means every time it is not a quasi-static process. It's a similar one, but not exactly same. Now you see what happens. If external pressure decreases, expansion or compression? If external pressure decreases, expansion or compression? Expansion, correct? Suppose this is a state A we have. We have some external pressure here. Now we decrease the external pressure. So we decrease the external pressure to some value, intermediate value, but once you decrease it, in this course we'll have some expansion like this, you see. This is the expansion we have in this, expansion. Further you decrease the pressure, we'll have again expansion here, like this expansion. Further you decrease the pressure, we'll have expansion again like this. And like this it goes, right? You keep on decreasing the pressure, we'll have expansion like this. Are you getting it? This is suppose we have a state B at this point. This is state B, A to B. So all these steps we are counting, this is infinite number of steps we have. Like this we have infinite number of steps. I have drawn this just to make you understand, but like this we have infinite number of steps possible, extremely slow process, and the entire process if you see, it goes like this from this pressure to this pressure. Like this it goes, correct? So pressure you see continuously changing from here to here, we'll have this expansion. Then again the pressure decreases, we'll have this expansion. Again pressure decreases, we'll have this expansion. Pressure decreases, we'll have this expansion. Clear any doubt in this? No, right? On this process, because we'll have the application of it, if you don't understand how to derive work done in isothermal reversible process, you won't understand it, right? Isothermal, you must have done work done in physics, no. Reversible isothermal expansion process, work done. What is the formula we have? W is equals to, could you tell me? Minus 2.303 NRT log V2 by V1, remember that? You must have done in physics, yes? So this work done, I'm just deviating from this topic a bit so that this is just to make you understand, you must have calculated the work done, the formula is 2.303 NRT log V2 by V1 in physics, correct? How do we derive it? Do you remember that? How do we derive it? How do we derive it? W is equals to yes, minus p external dv, isn't it? And then what we do? This we integrated dw, we integrated zero to w and we want, don't write this, okay, we want to V2. Now the question is, why don't you take this p external out of the integral sign? Could you answer this to me? Why don't you take this out? Why don't you take this p external out and devi just you integrate, why? Because p external is not constant for reversible process, remember that, right? So this you need to know. If the process is reversible, p external is not constant. And hence, we cannot take this out of the integral sign. You have to substitute it here, NRT by V and then you have to integrate it, correct? Right, that's why it is important to understand what is reversible, irreversible process. If it is irreversible process, we'll discuss irreversible now, then we can take this p external out because irreversible process p external is constant over there. And then directly the answer would be minus p external delta V. This is the work done we have in all irreversible process. Did you get it now? Yeah? Have you done reversible, irreversible process in physics? Okay, no problem. So this is the reason we have, why p external we took out of the integral sign? I can directly tell you, p external is a function of volume here. That's why we are not taking them out. But then we'll do this mother later on. Let's not focus on this formula. I'm trying to make you understand the importance of these two process because when it is irreversible, we have taken this out. But in reversible, we don't take this out. So reason is this, the one that I have explained, right? That is the first thing you need to understand. No, p external is not constant. That's what we are talking about. I think we're talking about expansion or compression. That is only possible when you change p external by some value. Yeah, if p external, we cannot say it is not constant. If the equilibrium is not reached, I'm discussing this irreversible, that's the second mother. See, if p external is not constant, then the piston would not be at equilibrium. It's not like that. Obviously, see, it's the hypothetical question Aditya. We must have some external pressure, right? So that pressure is constant only if you're not changing it. We are changing it out. It's not like it's happening on its own. We are putting the stone there. We are disturbing the equilibrium. We are increasing the external pressure. It is not happening on its own, correct? So it's not like p external is not constant than what happens. p external is always constant. External pressure is always constant. We are disturbing it, understood Aditya? Yeah, so this is one thing. So I hope you understood now the reversible process and why p external is not constant in irreversible, in reversible process. Now I'm going back to just make you understand the second process that is irreversible. In reversible, what happens, you see here? See, what happens here? Suppose this is one kg block we have a stone. I have put in, means total you see, I have put in four kg of block slowly. And slowly this piston will go down and down and reaches out to state B. Yes or no, I want you to respond here regularly because it is very important. Then only I will be understanding that you are getting it or not. See, so one one kg, just for reference I'm telling you, one one kg we have taken and slowly I put it here and then the piston will go down and will reach this state B. What happens, this is one way we can achieve this state. Why can't we put a four kg block here directly? Can we do that? Instead of one, one, one, one, one, I'll put total four kg of brick over there and piston will go down suddenly, right? Piston will go down suddenly and attains this equilibrium state. Can we do that? Assuming that it won't collapse, yeah. Yeah, piston is massless, correct. Piston mass we're not considering. Assume that this whole system will not collapse. Okay, it is wrong enough to sustain the weight of that brick. Okay, so instead of putting one, one, one, one kg block, I'll put four kg of that stone over there. The piston will suddenly go down, compression takes place and attains this state B. Can do that. So this process is faster than the previous one, right? The movement of piston you can observe. Piston will go down suddenly, right? All of a sudden, correct? This process is irreversible process. It happens fast, right? Against a constant, against a constant external pressure because the moment that you put in this block, heavy block of four kg, external pressure will rise to a certain value and compression takes place because of this new external pressure and that is same throughout the process from this point to this point, isn't it? So can we say here, the piston, sorry, the external pressure is constant? Can we say that? Did you understand? Tell me, did you understand all of you? You know, isobaric is constant pressure process. Okay, whether it is happening too fast or too slow, that's a different thing. Irreversible is different than isobaric. The only thing is common in irreversible and isobaric is constant external pressure, okay? Isobaric can be extremely fast or slow but reversible, irreversible is always fast process. Yeah, it is, see, that's what I'm defining. Try to understand the difference. It's not like four kg block is irreversible. Try to understand the difference in the two process, okay? State A to state B, we are going slowly, that is reversible. One other way is what? All of a sudden you increase the pressure and the pressure will go down to state B. That is one way also, which is faster than the previous one, extremely fast in fact. So this kind of process is irreversible process. Yes, so whatever the change, mother, we have, against that change only, the compression is taking place. It's not like I'm not talking about this pressure. Once you put the block over here, we have some external pressure plus some pressure we are putting in because of this block. So against this pressure, the compression is taking place. So whatever the pressure is, it is same throughout the process, state A to state B. Isn't it? No, we won't say the time is zero. We just say it is faster than the previous process. It happens so fast. Time thing we don't consider here. We don't define this in terms of time, okay? It takes a lot of time and it takes almost zero time, not like that. So did you understand the basic difference between the two process? Yes, this one, quickly guys. Basic difference you understood, correct? So this is why you see the external pressure in case of irreversible processes constant, but is not constant. But in case of reversible process, it is, it varies, right? It variable, constant for irreversible, variable for reversible, okay? So like we have seen some properties of reversible process, we have some properties of irreversible process also. I'm not going to draw the piston cylinder system over here. Just you write down few process and then we'll see the graph of this, okay? So it is first of all unidirectional. You cannot reverse it. That's why it is irreversible, unidirectional. Write down. We cannot reverse the process to the initial position. If you want to reverse it, you have to do some work on it, right? So work has to be done. Work has to be done on system, on system by surroundings, by surroundings to restore system, to restore system in its original position. Without work, it is not possible in its original position. Without work, it is not possible because it is irreversible, okay? Some, because the process is very fast, some amount of heat has been lost, some amount of energy has been lost, you know, in the form of heat. The process is fast, no. So because of friction or, you know, if friction you're not assuming there, if it is frictionless also, because of fast process, some amount of energy is lost in the form of heat. That's why if you remove the block, right? The energy that has gone already, you cannot restore that energy. So original pollution, you won't gain. If you want to gain, you have to provide some energy from outside. That is the work done. Understood? Yeah? So finite, so that's what you see. Fast process, right? There will be exchange of energy with surroundings. Slow process, the process is extremely slow, right? Exchange of energy with surroundings is not possible. That is also a very important point, okay? Next slide down, it has finite number of steps. Finite steps means what? We can count eight, 10, 12 like that. It's not like infinite steps, we cannot count that. So it has finite number of steps, fast process. They ask this in the school exam also, difference between irreversible and reversible process. It is the fast process, okay? It is real and actual one. The actual process is this only. The previous one, if you remember, we have discussed this imaginary, won't happen unpractically. System is at thermodynamic equilibrium. System is at, it's not like we don't have equilibrium over here. But the only difference is at thermodynamic equilibrium, thermodynamic equilibrium, at only initial and final step, initial and final step, at only initial at final step, correct? So we have equilibrium only at initial and final step. And that's why PV is equals to NRT we can apply only at initial and final step, right? On the next point, we can apply PV is equals to NRT, only at initial and final step, okay? And the most important one is we have P external constant, constant external pressures. One second, this copy is done. I'm just coming, I'll take some water and come. Just a second, yeah, copy this. Done? Yes. Done, no? Okay. Now, if you look at the graph here, graph for this process, okay? So we have pressure, volume. External pressure is this initially when state A we have external pressure is this. Now, when you put the block, 4KG block, all of a sudden the pressure will come down, right? It is state B. And in this process, what happens? The expansion takes place, right? And the graph will go like this. Expansion goes like this. Okay, it won't touch the, you know, x axis here. We'll just remove this point, just to make you understand I have taken this graph. So this is a decrease in pressure and in that pressure the expansion takes place like this. So constant pressure, which is this. Against this pressure only the expansion is taking place. So these are the processes you need to understand and keep in mind, okay? No, this is just here to make you understand. It's like pressure was here initially. Once you increase the pressure, it will go down. And then the pressure volume expansion happens. No, so it goes like this. So it's a continuous graph.  There's no steps over it's continuous graph like this because both happens, you know, the pressure increases all of a sudden, decreases all of a sudden and then the expansion takes place like this. I have taken this, the example of irreversible expansion. You will show this graph to what they see. It's not like these two points you will show, right? The relation of PV is like this only exponential relation. So you draw the graph like this. The expansion is why the graph stops at this point because it's happening against this pressure. That's why it stops over here. No, PV is goes to NRT is applicable only at this and this point, initial and final. And it's part does not matter over here but it goes like this only. So this is the processes you need to understand, okay? Now next slide down, we'll see thermodynamic quantity. After finishing this thermodynamic quantity, we can think of, you know, the first law of thermodynamics we can start and then we can start the derivation also or work on another things, correct?  First one is work. Work is generally we have PV work we are discussing. Pressure volume work, okay? So for this first of all, we have an IUPAC convention. This you need to memorize, IUPAC convention. And what is this convention? On the system is always positive. Just a second. Work done on the system is always positive. Whenever work is done on the system, it is a case of compression, right? Work done by the system. System is doing work, right? By the system is negative. Remember it is Ulta in physics, okay? It is a case of expansion. So there's no any mistakes over here. This is the convention of we use in chemistry. In physics it is opposite, okay? The work done formula that we have, general formula is DW is equals to minus of P external into DV. This is the general formula we have for all processes. The expression of work done, we always get from this particular formula. Okay, all the processes will just apply the condition of a particular process and we'll get the expression of work done from there, okay? So this is the relation we have. Now in this, if you find out W, so we have to integrate it, V1 to V2 and zero to W, correct? So W is equals to we get minus P external DV V1 to V2. This is the in general expression we have, which we use to find out the expression of work done various process. Yes, understood. The unit of work done is we can have, yeah, unit of work done we can have, Joule we can have Caledee, we can have ATM liter, and we can have, yeah, we can have ATM liter, okay, anything we can have. Whenever you do P into V, so pressure is an ATM, volume is in liter, so mostly you will get the work done in ATM liter. So always keep that in mind. So whenever work done is an ATM liter, you have to convert it in Joule. So one ATM liter, one ATM liter is equals to 101.325 Joule, with this you can easily convert. Or if you don't remember this, you must know that 0.0821 R value, 0.0821 ATM liter per mole Kelvin, right? ATM liter per mole Kelvin, no, no, depends on the option, but I would suggest take this value only. We don't approximate it to zero. And we also know R value 8.314 Joule per mole Kelvin, right? So if you see these two value here, can we write this 0.0821 ATM liter is equals to 8.314 Joule? So from this you can always find out one ATM liter equals to 8.314 divided by 0.0821 Joule. So with this expression also you can find out the relation. Got it? So any one, you know, relation we can keep in mind. So this is the work done thing, right? Now we have few conditions into this and in different, different processes, how to find out work done. Okay, we'll continue with this in the next class. But after next class you can start solving numericals on this. Okay, okay, thank you. Take care.


Law without government. Part 1. Principles. What is government? Government is defined as a territorial monopolist in the field of producing law. It is the sole provider of law, the ultimate decision-maker, arbitrator and wielder of force within a territory. As a monopoly, it maintains its position by using aggression, the use or threat of violence, to prevent competing providers of law from emerging. Government is the only organisation that uses the political means, that is the widely accepted use of aggression to attain wealth. For example, the monopolist declares its own act of theft to be legal, calling it taxation and enforcing compliance. Everyone else must attain wealth using the economic means, producing something of value to others and then engaging in voluntary acts of trade. Government is a territorial monopolist of law, but what is law? Interpersonal conflicts are possible due to material scarcity of resources and goods and diversity of interests between individuals. The potential for conflicts makes property rules and ownership rights necessary for social cooperation. For example, apples are scarce and this means that if two people both want to eat a particular apple, they cannot both be satisfied. For conflict avoidance, we need property rules to establish who has the ownership right over the apple, that is who has the right to decide how the apple is used. Laws are property rules that emerge from the resolution of conflicts. The production of law, the resolving of conflicts, is a service provided by an arbitrator or judge. Imagine two individuals stranded on a desert island, Adam and Ben. Adam picks a supply of apples, but then Ben comes along and takes an apple without Adam's consent. That's my apple because I picked it. It's my apple because it was on my tree. With no one else on the island, Adam and Ben have no one they can turn to for help resolving this conflict. They may succeed in negotiating a peaceful settlement or they may resort to physical violence. Now suppose there is a third individual on the island, Charlie. Now there is another possible way for Adam and Ben to resolve the apple conflict peacefully. Ask Charlie for his opinion and agree to whatever resolution he suggests. We're having a dispute over an apple. Both of us claim it as our own. Will you arbitrate for us? This is third-party dispute resolution. Adam and Ben both make their cases to Charlie. Charlie must decide who he believes as the stronger claim to the disputed apple and then pronounce a judgment on the case. I do not think Ben owned the tree, so I award ownership of the apple to Adam. Charlie has just produced a law. He's made a judgment about who the rightful owner of a disputed property is. He is awarded legal ownership of a property to one of the disputants. But Adam feels that it would be unjust if Ben only has to return the apple he stole. Adam wants Ben to be punished and wants compensation for having his time wasted. He insists that Ben pay him five additional apples and then he will consider the matter settled. Unable to resolve this dispute between themselves they asked Charlie for his opinion. Charlie recognises the need to compensate Adam for his lost time and to punish Ben. His opinion is that a payment of two additional apples from Ben to Adam would be a just resolution to this conflict. Since the purpose of Adam and Ben turning to Charlie was to help them resolve the dispute peacefully, both men will agree to his decision. If one of them does not, then they are back to having to resolve the conflict between themselves, either peacefully or otherwise. By arbitrating on a conflict and helping to resolve it peacefully, Charlie has produced a law. Now suppose some time later on the island another conflict occurs, this time between Adam and Charlie. If they cannot resolve the dispute between themselves, they could ask Ben to arbitrate for them. I'll arbitrate for you. When Ben provides them with his opinion on the conflict and suggests a resolution, he too will have produced a law. And if Ben and Charlie ever get into a dispute, they could ask Adam to produce a law for them. There are multiple producers of law in this society. No single producer of law is in a privileged position. There is no ruler and no one is ruled. Everyone is of equal status with respect to the laws. What would a monopoly of the production of law look like on our island? I'll arbitrate for you. No. You are not allowed to arbitrate. I am the only one who can produce law on this island. My law is the law. The injustice of this arrangement would be immediately apparent to both Adam and Ben. But that would mean that you even get to be judge in disputes you are involved in.  That's right. I am the state. Charlie could only establish himself as ruler and maintain that position if he could somehow convince Adam and Ben that a ruler is necessary and that with no ruler anarchy there would be chaos and disorder. If Charlie is able to maintain a monopoly of arbitration and ultimate decision making he would have put himself above the law. And Adam and Ben can no longer be considered free men. Now suppose there are a few more individuals in this island society and two of them have a dispute that they cannot resolve peacefully among themselves. The disputants have a choice of arbitrators that might help them resolve the conflict. There is competition in the production of law. Who will they choose? The ideal arbitrator will be someone who is impartial and who has a good reputation for being fair, honest and wise. With a larger population some individuals who possess these qualities may find that they can make a living purely by providing arbitration services to disputants. They will be professional judges and may create firm selling laws. Their consumers will be disputants who need help resolving a conflict and their income will depend on their reputation for making wise and fair decisions. If any one of them tries to become a monopolist for example by insisting on being judge in a case involving himself or a member of his family he will quickly lose his reputation and his livelihood. The principles of having competition in the field of law do not change as society becomes larger and more complex. In my next video Law Without Government Part 2 I apply the principles outlined here to a large and complex society explaining how law could be provided by competing firms.


Hello and welcome to the next lecture in the course on Introduction to Computer and Network System Performance. My name is Varsha Apte and I am a faculty member in the Department of Computer Science and Engineering. So, in today's lecture, we will continue what we were doing in the last lecture which is we were discussing high and low load asymptotes for various metrics of the various standard queuing systems that we are now learning, okay. So, here is again the slides that I show now to remind you of all the parameters of our queuing systems. All the analysis that we are doing is in the, in terms of these parameters, the arrival rate, the service time, the service rate which is 1 over this service time and we have the number of servers and we have the buffer size and the metrics are always these, the number of jobs in the system, the number of jobs in the queue, utilization, this which is also the fraction of time the server is busy throughput, time spent in queue, time spent in queue and service and blocking probability, right. So, these are all our standard things that we are doing. Recall that this is actually we call this utilization and this is throughput. Also we use jobs and requests and customers kind of interchangeably for a general queuing system. And this is something we learnt in the so far about the metrics and some laws. So, this is also this also lists in a clear list the queuing systems that we have been learning so far GG1, GG1K, GGC, GGCK, these are the 4 queuing systems we are learning. Actually these 2 are just special cases of this, this is nothing but when you have C equal to 1 and here GG1K is also again when you have C equal to 1 and this is when you have K equal to infinity and this is both C equal to 1 and K equal to infinity. So, these are just actually special cases of these queues, but it is just useful to list them separately. So, that is why we have done that. You can go back to the previous lectures to see how we derived all of this out of this particular the utilization formulae come from utilization law. So, this is just a recap. Last recap is of the previous lecture where we basically derived asymptotic values. So, high and low load what we mean by that is high load is when the arrival rate is going to infinity, low load is when arrival rate is almost 0. So, what kind of metrics can we expect for that this lists that out. So, remember that throughput will go to the maximum capacity in both cases utilization of course, server will become busy if it is highly loaded in the only main differences is in the finite buffer queue. The number in system and the response times actually converts to some finite values whereas the number in system and queue lengths and response times and waiting times for the infinite buffer system as expected they just go to infinity. Another thing to remember always is that even with very little load response time is always about a customer that actually get service. So, this is for those who get service it is also to those who get service. So, if you are going to get service if you are going to be admitted into the system and get service then you will certainly need a minimum amount of response time of tau. So, that is the thing of course, for infinite buffer we have no loss probability whereas in finite buffer for high load loss probability is going to go to 1. So, this is just to recall what happened so far. Let us move on now just straight forward generalizations of whatever we did earlier for GGC and GGCK. So, out of that the GGC is actually quite simple let us make the same table as before we have our throughput utilization number in queue number in system waiting time response time and lambda tending to 0 lambda tending to infinity. So, again same thing here the really now you should be able to do this yourself I would encourage you to pause the video and actually try to fill this table yourself ok. So, you can look at the previous slides how the reasoning has been and you should be able to easily fill this table yourself. So, of course, when there is very little load coming to the system throughput also will be going to 0 utilization is 0 all of these are 0 except always remember that response time is tau when the load arriving into the system that is arrival rate into the system is increasing to infinity. Now, remember that if there are C servers each going at rate mu remember that mu is equal to 1 by tau this is going to go to C mu and utilization will go to 1 all of these again we have infinite buffer remember when you do not have anything written there it is infinite buffer. So, these are just going to go to infinity. So, really the only difference between gg1 and this was that this maximum throughput that as lambda tends to infinity throughput will go to C mu instead of mu that was the only difference. What is more interesting is actually this ggck ok. So, let us first do the throughput utilization and both the queue length and also number in system. So, let us draw the same table so throughput utilization number in queue number in system waiting time response time lambda going to 0 lambda going to infinity. This should follow the similar reasoning as gg1k so if you want to look at the slides and just pause the video here and try to fill this yourself it will be it will be a good exercise. As usual let us do the easy part first let us do the lambda tending to 0 first throughput of course will be 0 we do not expect the server utilization to be much it will it will also go to 0 as lambda goes to 0 all of these this is really no difference always this is tau no matter if there are C servers remember the customer once they say it starts service it will anyway need tau amount of time for the work that it has to do. So, this will remain tau ok now let us look at what happens at as arrival rate increases. So, this will go to C mu throughput is going to go to C mu server will become fully busy queue length remember we have k size buffer and these are C servers. So, the number that can sit in the queue as lambda tends to infinity you expect this to become full ok. So, clearly this will be k as lambda tends to infinity the number in system is this k plus this C right. So, this becomes k plus C ok. So, these 3 plus even the number in system all of this is kind of straightforward the waiting time and response time asymptotes are a little bit interesting. So, let us take another page for that ok. So, let us try. So, what are we trying to find here lambda tending to infinity what happens to waiting time and what happens to response time ok. So, let us draw the queue again here this is the this is k this is C to understand this again it is really much better to take an example. So, let us take an example of a queuing system with size 3 in the buffer and maybe 2 servers here ok 2. So, we have k is equal to 3 and C is equal to 2 and let us take the tau how about the tau as 10 milliseconds ok. Now, the question is as lambda tends to infinity what happens to the let us try to first do the waiting time ok. So, as lambda tends to infinity remember that the queue will pretty much always be full ok. So, actually we expect even as the servers to be fully busy and the queue to be full what will happen just when a departure happens ok just when a departure happens let us say this server some departure will happen and the request that is over here will go here the request that is here will go here and the request that will go here. So, we expect that the new arrival will always be the last one here ok it will always be the last one. It is very very low probability that a new arrival even if they get space in the queuing system that it will actually be able to queue anywhere like you know that it can be the first or second that probability is almost 0 with probability 1 a new arrival will be the last right I hope that is intuitive that if there are so many the load is so high that as soon as even one space is open at the end of the queue the new arrival will be there ok. So, we basically start with this assumption that any new arrival will be last in queue. So, since it is the last in queue how long will it may take for it to go from here to here that is the question we are trying to do ok. So, let us look at that ok. So, again I am here I will move from here to here when the request in service here finishes its service and remember that I would have joined just when a departure has happened. So, I will wait for here to here for a full service time ok an entire tau you know actually sorry. So, so with probability 1 remember I will be queuing the last ok. Now at what rate am I going to move through the system ok this is something very important to remember ok. So, let me actually draw this thing again here. So, at what rate are requests leaving remember that when the load is very high all these servers are very busy they are continuously busy and we expect that a request will be appearing on the on this side that means they will depart once every 10 by 2 milliseconds ok. Why is that? Because from this server you can expect from this server that requests leave every 10 milliseconds and from this server you will expect that they leave every 10 milliseconds. So, if you look at the merged stream here ok if you look at the merged stream here you can expect that on an average you will see once a departure once every 5 milliseconds right that is intuitive right. If I am seeing one departure here one request finishing from this server every 10 milliseconds from this server every 10 milliseconds then as a total as an aggregate departure rate here I should see one every 5 milliseconds. What does that mean? Why am I doing all this calculation? Because remember that this queue is full here and you can move through this queue only at the same rate at which departures are happening through this server ok. So, for the first at the first 5 milliseconds you will need one 5 milliseconds to move from here to here which is when this customer will start service. So, this is 5 milliseconds then you will need another 5 milliseconds to move from here to here this is 5 milliseconds and remember now you are still at the head of the queue you will need one more to move into the system ok. So, you needed 3 into 10 by 2 this many milliseconds you needed to move through the queue and start service and this was your waiting time. So, in general can we generalize this? If we generalize this it means a customer that stands at the end of the queue has to move through k positions it has to move k times and this movement happens each movement takes how much time in general here tau was 10 milliseconds and we had to divide tau by 2 because number of servers were 2 in general it will take tau by c milliseconds right. So, waiting time is going to be equal to k multiplied by tau by c as lambda tends to infinity right. Now, once we have the waiting time response time is trivial take response time is equal to waiting time plus tau. So, this is going to be equal to k tau by c plus tau. So, that is it with this ggck asymptotes let us look at them everything together for ggck of course, low load everything is 0 except response time and the throughputs all of these the throughput for ggc and ggck high load asymptotes are going to be the same they so utilizations also high load asymptotes are going to be the same. For number in queue and waiting time and all of these things the infinite buffer queue goes to infinity. The finite buffer queue k is easy to know that this is we go to k total number in system will go to k plus c waiting time we just saw the whole reasoning that it goes to k tau by c and response time is nothing but waiting time plus service time. Loss probability is one more very important metric for infinite buffer it remains 0 there is no loss for finite buffer at low load it will be nearly 0 and at high load it will be nearly 1. This and the fact that the queue is full this is related that nearly all of the requests that come in are going to be dropped. But for those that are not dropped and somehow make it to the system these are the limiting this is the limiting value of asymptotic value of waiting time and this is of response time. So with that let us now just do some examples. We have done a lot of formulae and theories so we are just going to go through some examples. This is the first example is of a web server it is a very common example. Let us say there are two threads running on two cores in a CPU this is a server machine. And for all our examples let us assume that the threads are only using CPU and for the beginning let us assume it is an infinite buffer. So what we have is a gg2 queuing system let us look at the asymptotic values all the if nothing is coming to the system then through port utilization all of these are 0 response time will be equal to service time so that is 10 milliseconds. What is the throughput? So for that we have to calculate c mu so we have c equal to 2 here and mu equal to 1 by 10 in terms of requests per millisecond. So this is going to be 1000 by 10 requests per second. We can leave this is also correct it is just a convention to always mention request rate in terms of requests per second. It is a little easier to look at that number 100 requests per second. So that is what we have for mu and then we multiply this by c so c mu is equal to 200 requests per second. So that is how we get this and then we have the utilization as 1 in the infinite load high load asymptote and again this is infinite buffer so all of these become infinite. Now let us look at the same example but with a finite buffer. Note that it is the same example 2 threads running on 2 cores request processing time is 10 milliseconds but there is a queue size now that is you can have 10 here and we have 2 cores here c is equal to 2, tau is equal to 10 milliseconds. Our mu does not change remember we calculated mu will be 100 requests per second and c mu also is the same. What else do we have so all this now is the same remember the low load asymptotes really do not change this is 10 milliseconds response time is the only one that has a non-zero value everything else is 0 at infinite load however lot of things change right. Of course throughput and utilization are the same as the gg2 infinity queue but number in queue clearly is going to be 10 because that is the queue length the buffer size number in system will be 10 here and 2 here so we have 12. Waiting time remember the exit rate here from here we are going to have 1 every 10 milliseconds from here we are going to have 1 every 10 milliseconds the aggregate rate will be 1 every 5 milliseconds so the stepping up the request will be able to step up through this queue at the rate of once every 5 milliseconds and you have to move 10 positions right. So remember all the reasoning we did earlier so we will have 10 multiplied by 5 and that is equal to 50 milliseconds that is where the waiting time comes from and then we have response time as just 50 plus 10. Next example this is interesting this is the cellular channels example suppose a cell of a particular operator has 10 channels and the average call holding time that is the average amount of time that somebody calls talks on the phone for 1 call is 2 minutes and remember that cellular the way cellular calls operate is the buffer size is 0 there is no such thing as queuing a call calls that arrive if all the channels are busy then you just get that network busy signal and the call is dropped right. So we have here a gg2 0 system sorry gg this should be 10 gg10 0 system. So now how does the rest of the calculation proceed remember the call holding time is 2 minutes so in one we are this is again another convention that the throughput for calls is usually done in a per hour rate. So if one call it takes 2 minutes in 60 minutes which is 1 hour we can do 30 calls right and now remember that there are 10 channel this is basically this is our tau in minutes this is our mu in calls per hour and our C is 10. So our C mu is going to be 300 calls per hour right that is how we get 300 as a high load asymptote then utilization of course all the channels are going to be busy if there is a lot of load number in system is going to be 10 remember there is no buffer here. So it is just going to be 10 as calls go to infinity call arrival rate goes to infinity all there will be always 10 calls going on because there are 10 channels in the system. Waiting time is 0 because if a call is admitted into the system it is immediately admitted it immediately gets a channel. So waiting time is always 0 because there is nowhere to wait if you are in the system means your call has started. Response time of course is going to be the service time so 2 minutes. So here it is interesting the low load asymptote and high load asymptote of response time is the same and that is because there is no buffer. You can visualize this system as like this there is no q there is no q k is equal to 0 and last probability of course at low load is 0 and at high load almost all calls are getting dropped. Now a few non asymptotic metric we have only done asymptotes in this first 3 example let us just do some usual metrics along with the asymptotes. So let us go back to a single threaded server running on a one core CPU. Let us start with infinite buffer request processing time is 5 milliseconds and request inter arrival time is 6 milliseconds. That means mu will be 1000 by 5 which is 200 requests per second and lambda is going to be 1000 by 6 which is around 167 requests per second. Yeah the asymptotes now you can kind of look at it yourself everything for lambda going to 0 other than response time which is equal to service time everything is 0. So this is infinite buffer so loss probability remains 0 all this number and waiting times are infinity this is our C mu and CPU utilization given as percent is going to be 100 percent. Now but now what is interesting is to actually do this at the given lambda which is 1000 by 6. So this is not lambda going to infinity. So throughput is going to be 167 why because 167 is less than 200 because arrival rate is less than service rate. CPU utilization remember it is lambda tau which is equal to lambda by mu so which is also equal to you can write it as 1 by mu by 1 by lambda which is equal to tau over 1 by lambda. So in our case remember tau is equal to 5 milliseconds and 1 by lambda is 6 milliseconds. So you can write it as 5 by 6 and just to make it a percent you get 83 percent from that. And then we do not have what we do not know here is that at a lambda which is neither going to 0 nor going to infinity what is the queue length what is the number in the system what is the waiting time what is the response time we do not know that we also do not know the loss probability actually we do not know this. Now the same example but for limited buffer again the asymptotes I can I will encourage you to look at it yourself we did a very similar one earlier we did it with two cores this is with one core and with 5 milliseconds response time these remain the same 10 is the buffer size buffer size plus 1 and 50 milliseconds is the waiting time because we have to move 10 positions and 50 milliseconds plus 5 we get the response time. What is interesting here now is that the throughput we know is going to be arrival rate into probability of the request being accepted into the system. So, which is 1 minus loss probability. Now loss probability is something we do not know that is because at a low load asymptote we can assume it to be 0 at a high load asymptote we can assume that it is going to 1 but at a low load asymptote we still do not know how to calculate the loss probability. So, we have to just leave this as this equation and all we can say about the throughput is that it is something less than 167 requests per second. All we can say is that it is something less than arrival rate because there is going to be some nonzero drops happening and therefore, all the incoming requests are not going to be accepted and therefore, the throughput is not going to be equal to arrival rate it is going to be something slightly less than the arrival rate. Similarly, all that we can say about the utilization is that whatever we had calculated earlier we have we know that it has to be less than that 83 percent because throughput is not exactly equal to 167 it is something less. So, we have to leave it at that we also do not know how to calculate queue length, numbering system, waiting time and response time at this value which is neither 0 nor infinity. Now, let us do the same example for a two threaded server on dual core CPU. The asymptotes change here this is mu was 200 requests per second but now we have c equal to 2. So, accordingly this is 400 of course utilization will always go to 100 percent and queue length assumptions here the same this is 10, this is 12. This changes however because now we have two CPUs and we are going to be moving through these 10 positions in a 5 by 2 milliseconds per position. So, that gives us 25 this comes from 10 multiplied by 5 by 2 and now we get 25 milliseconds as the waiting time to which we have to add 5 milliseconds as the service time. So, that is how we get response time equal to 30. And now we have loss probability as usual should go to 1 when lambda 2 tends to infinity. What about the intermediate things at 1000 by 6 actually throughput this remains below 400 but it is not going to be exactly 167 we know that it is a limited buffer. So, there will be some loss. So, from the system some are getting dropped here with probability P L and only lambda multiplied by 1 minus P L are making it into this buffer and therefore those are the ones that leave the system. Those are the ones that are going to leave the system. So, that is the throughput that remains the same since 167 remained below 400 the same formula applies. But now we have all of the numbers for the same the only difference was that there are two servers. So, actually it is the same thing but it gets divided by 2. So, in percentage earlier it was around 83 now it is around 41 percent. So, this is just summarizing everything you can go through these formulae we have derived all of this in all of the previous lectures. And in fact what I would like to point out is gg1, ggc and gg1k are all special cases of this you can either set c equal to 1 or k equal to infinity and you get all of these special cases. The point here that I want to make even here in fact I want to just show you that how does this become let us say how does this work for an infinite buffer for infinite buffer pl will be equal to 0. So, actually this will just become this will become 1. So, this becomes small lambda. So, this is basically the formulae they get basically specialized for each of these cases. So, you can see here that all of these values are filled for the asymptotes, but for the not asymptote only this first two values we have some expression for. And even that we do not actually for the infinite for the finite buffer condition we actually do not know what the throughputs are for knowing the throughputs and the utilizations also we need to know the lost probability. And in the infinite buffer case we actually do not know these values at all for even for anything that is not any lambda that is not 0 or infinity. So, how do we find this? So, for this we actually need a different and more advanced theory if we really want to find all of these values we need something called stochastic processes. But luckily there is another simpler law which is called Little's law which helps us find let us say the number in system if we are given the response time or vice versa or if we have the queue length then we can get the waiting time or if we have the waiting time we can get the queue length. So, this is what we are going to do in the next class.