1 00:00:13,809 --> 00:00:23,779 Okay, so now we will actually look into how to differentiate between probability of call 2 00:00:23,779 --> 00:00:29,141 loss and probability of switch being in blocked state these are two different things. And 3 00:00:29,141 --> 00:00:32,419 the ways of actually estimating these two in the switches. 4 00:00:32,419 --> 00:00:33,910 Unable to capture the image because the lecturer writes and erases the content immediately. 5 00:00:33,910 --> 00:00:43,870 So if there is a switch and there is a call which arrives and when the call arrives it 6 00:00:43,870 --> 00:00:50,309 finds that switch is in the blocked state. Now call will not be able to go through, so 7 00:00:50,309 --> 00:00:55,629 the probability that your call will not go through is known as call loss because call 8 00:00:55,629 --> 00:01:02,540 has been lost. So if you make 1 million calls out of which 10,000 calls you found that switch 9 00:01:02,540 --> 00:01:07,030 was in block the state at that point of time call cannot go through. 10 00:01:07,030 --> 00:01:19,340 So that is a call loss or we call it a call congestion okay. Secondly even if the call 11 00:01:19,340 --> 00:01:24,530 comes or call does not come does not matter you look whether switch in which states, if 12 00:01:24,530 --> 00:01:28,999 switch is in the blocked state even if call does not come it does not matter if there 13 00:01:28,999 --> 00:01:33,530 is no call coming in switch is in blocked state there is no loss, there is no call is 14 00:01:33,530 --> 00:01:41,270 being lost actually in that case. So the second possibility is we call it a, that switch is 15 00:01:41,270 --> 00:01:52,130 in the blocked state, so that we callas a time congestion. 16 00:01:52,130 --> 00:01:57,450 So now question is how these two are going to be related, one thing which is sure that 17 00:01:57,450 --> 00:02:04,209 then switch is in the blocked state that is basically fraction of you are actually technically 18 00:02:04,209 --> 00:02:09,700 observing the time when for example, I estimated in my previous lecture probability of switch 19 00:02:09,700 --> 00:02:17,140 being in blocked state what does that mean, you observe the switch for a long time say 20 00:02:17,140 --> 00:02:22,220 a few years and you find out fraction of time for which switch was in that state which is 21 00:02:22,220 --> 00:02:25,490 the blocked state. So ratio of that particular time divided by 22 00:02:25,490 --> 00:02:31,760 the total observed time will give you the time congestion that is why the time congestion 23 00:02:31,760 --> 00:02:36,600 word is being used, but the probability that your switch is in blocked state does not mean 24 00:02:36,600 --> 00:02:42,770 call is blocked, only when the call will arrive then only the blocking will happen for the 25 00:02:42,770 --> 00:02:50,840 calls. So call congestion usually will be lower value having a lower value then the 26 00:02:50,840 --> 00:02:54,959 time congestion. So that I think intuitively is clear, but 27 00:02:54,959 --> 00:03:02,860 let us build up a relationship between these two. So if we define that probability that 28 00:03:02,860 --> 00:03:13,090 of call arrival is defined as PA probability that a call is arriving and I also define 29 00:03:13,090 --> 00:03:30,290 PN(a) this is the conditional probability 30 00:03:30,290 --> 00:03:53,900 that call arrives when switch was in Blocked state 31 00:03:53,900 --> 00:04:05,200 okay so I define call congestion as probability of call loss PL I define the time congestion 32 00:04:05,200 --> 00:04:11,700 as PB being in the blocked state. So now I actually have four variables and 33 00:04:11,700 --> 00:04:16,770 these need to be related to each other, remember this is a conditional probability conditional 34 00:04:16,770 --> 00:04:23,010 probability when switch was in a blocked state and a call arrives, so this call is going 35 00:04:23,010 --> 00:04:30,770 to be lost and when it probability that a call arrives this is the probability that 36 00:04:30,770 --> 00:04:41,919 the call will be loosed okay condition on that call arrives actually so P x L into P(a) 37 00:04:41,919 --> 00:04:48,200 this gives the number of calls which will be getting lost over time okay remember when 38 00:04:48,200 --> 00:04:52,140 I am saying when a call arrives and it finds switches in blocking state that is a conditional 39 00:04:52,140 --> 00:04:55,639 probability. That is a call congestion and this will be 40 00:04:55,639 --> 00:05:00,949 on an average number of calls lost over long time if my arrival rate is high more number 41 00:05:00,949 --> 00:05:07,010 of calls will be lost arrival rate is lower less will be lost over time okay, so this 42 00:05:07,010 --> 00:05:12,370 should be equal to because both are going to lead to the same thing this will lead to 43 00:05:12,370 --> 00:05:18,380 nothing but probability that is a conditional probability that call will arrive and switches 44 00:05:18,380 --> 00:05:25,470 in block the state and probability that you will be in block the state is PB this technically 45 00:05:25,470 --> 00:05:29,960 is a Base rule. Okay what I am saying is that they are event 46 00:05:29,960 --> 00:05:36,330 and B switches in block the state and calls arrive so this P(A,B) be that both things 47 00:05:36,330 --> 00:05:47,590 I happen is P (A|B) so probability that switches in block B state probability that call arrives 48 00:05:47,590 --> 00:05:55,360 condition none switches in blocked state okay so this represents right side it is also possible 49 00:05:55,360 --> 00:06:08,860 to write P(B| A) the call arrives and when the call is arriving on that condition that 50 00:06:08,860 --> 00:06:13,470 is which was in block the state at that point of time, is a technically basil which I have 51 00:06:13,470 --> 00:06:19,180 written here and this is what I am going to use to estimate the relationship between PL 52 00:06:19,180 --> 00:06:32,310 and PB. So I can now write and we will do it for m/m composites which only. 53 00:06:32,310 --> 00:06:52,340 I can right now PL(S) and of course now very important observation which you can make, 54 00:06:52,340 --> 00:06:59,169 probability of call loss and probability of blocking will be almost same, if my call arrivals 55 00:06:59,169 --> 00:07:04,539 probabilities are independent of switch state if they are independent of switches state 56 00:07:04,539 --> 00:07:10,110 this and this has to be exactly same and they will cancel both of them will be called loss 57 00:07:10,110 --> 00:07:16,120 probability or call congestion and time condition will be same, okay. 58 00:07:16,120 --> 00:07:22,569 But that actually does not happen this is going to be a true statement if my n is comparatively 59 00:07:22,569 --> 00:07:28,569 much smaller than m, m is very large. So the arrival probabilities does not get impacted 60 00:07:28,569 --> 00:07:34,819 by the change in switch state they will be almost same and both of them will be similar 61 00:07:34,819 --> 00:07:44,319 but when m and n are not n is not much smaller not much smaller than m then this is a PL 62 00:07:44,319 --> 00:07:51,759 = PB cannot be taken we have to actually estimate a relationship between that. 63 00:07:51,759 --> 00:08:00,319 So what is PN or K let me see how we can find out, so when the switch is in state A then 64 00:08:00,319 --> 00:08:09,450 what is the arrival rate, I have to find out that and the small elemental time ??T so I 65 00:08:09,450 --> 00:08:21,210 can put some ??T here. So that is a probability of arrival when you are in state n in small 66 00:08:21,210 --> 00:08:37,140 time ??T, probability of arrival this will be the average value of the arrival rate, 67 00:08:37,140 --> 00:08:41,260 with every state change there is going to be different arrival date. 68 00:08:41,260 --> 00:08:46,920 Remember in the previous lecture when you were in state zero, arrival rate was M? when 69 00:08:46,920 --> 00:08:54,070 you are in state one it was M -1? and so on, I make an estimate average estimate depending 70 00:08:54,070 --> 00:09:00,800 on that probability of the events and I will get this average value in totality, so that 71 00:09:00,800 --> 00:09:04,570 is what is going to be the call arrival rate on an average basis which is independent of 72 00:09:04,570 --> 00:09:15,360 state states actually have been ironed out by taking what we call a proud mystic average. 73 00:09:15,360 --> 00:09:24,700 So in this case ?T now can be estimated as because I am taking a probabilistic average 74 00:09:24,700 --> 00:09:31,390 i can write that you I am in probability of being in state k & state k what is the probability 75 00:09:31,390 --> 00:09:39,740 of what is the arrival rate so I am taking a probabilistic average am I take all states 76 00:09:39,740 --> 00:09:53,010 goes from k to n so that will be PA. So I can now solve it so i have already remember 77 00:09:53,010 --> 00:10:04,060 I am taking ratio ??T is will cancel these are immaterial so PL will be ? n divided by 78 00:10:04,060 --> 00:10:17,910 ? of I have to write down this thing 79 00:10:17,910 --> 00:10:31,700 okay so this is what will be the relation now ?T there I will rate and let the end they 80 00:10:31,700 --> 00:10:41,399 will be related ?n will always be smaller than ?T okay this of course you can make a 81 00:10:41,399 --> 00:10:53,410 drawing so when you were in state 0 m? was the arrival rate as you keep on moving by 82 00:10:53,410 --> 00:11:05,370 state it becomes a - 1 number and you will keep on reducing at this point it will be 83 00:11:05,370 --> 00:11:10,220 M -n ?. And this monotonically decreasing as your 84 00:11:10,220 --> 00:11:21,610 state’s as you are actually going to from 0 to nth state so the average value will be 85 00:11:21,610 --> 00:11:28,010 somewhere here and there the smallest value so this top one is the smallest value so this 86 00:11:28,010 --> 00:11:36,959 has to be smaller than the average and of course with this you configure out that this 87 00:11:36,959 --> 00:11:43,030 is going to be smaller than this so PL is always going to be less than PB this was also 88 00:11:43,030 --> 00:11:50,510 the intuitively the result which I explained in the beginning the call loss probability 89 00:11:50,510 --> 00:11:57,639 will be smaller than the switch being in the block state that probability from okay it 90 00:11:57,639 --> 00:12:06,790 will be smaller than that so let us solve it so ?n essentially will be M -n so I am 91 00:12:06,790 --> 00:12:08,709 just going to put everything in this expression. 92 00:12:08,709 --> 00:12:11,980 Unable to capture the image because lecturer writes and erases the content immediately. 93 00:12:11,980 --> 00:12:38,940 And we will solve it and this expression will be 94 00:12:38,940 --> 00:12:45,389 that is your ?k probability that you are in that state I am going to use the Tang set 95 00:12:45,389 --> 00:13:09,829 probity distribution will be given by MC k ?/µk /again there has to be a summation 96 00:13:09,829 --> 00:13:25,430 okay so I can also now to place this PB in the blocking probability x in fact I have 97 00:13:25,430 --> 00:13:37,899 to put again now this is independent of k this and these two are same so I can cancel 98 00:13:37,899 --> 00:14:04,899 them actually once I cancel I will get I can actually raise it now from here okay. 99 00:14:04,899 --> 00:14:20,110 So this ? will cancel with this one this M factorial 100 00:14:20,110 --> 00:14:56,620 by n factorial m - n m factorial by K factorial so 101 00:14:56,620 --> 00:15:09,509 I can cancel and I can make it - 1 so and I can make m here and take 1 m here out i 102 00:15:09,509 --> 00:15:37,130 can do the similar exercise here this am will cancel and i will have 103 00:15:37,130 --> 00:15:54,550 M -1 C n ? by µ n ? k goes from 0 to n, m - 1 CK and of course you can see this is the 104 00:15:54,550 --> 00:16:03,089 same expression as the probability of being in blocked state for M by n composites which 105 00:16:03,089 --> 00:16:06,839 except now instead of m I am going to have M -1 here. 106 00:16:06,839 --> 00:16:16,910 So i can write this thing as nothing but probability of blocking but for m - 1 inputs instead of 107 00:16:16,910 --> 00:16:23,090 using M by n if I use M -1 by n composites which the probability of it being in block 108 00:16:23,090 --> 00:16:29,220 the state is equal to probability of call loss for M by n switch so which actually implies 109 00:16:29,220 --> 00:16:40,190 that p l of m should be equal to PB (M -1) that is how your call congestion and time 110 00:16:40,190 --> 00:17:16,720 congestions will be related to each other.