Lecture Notes of Communication Network I y: part 5. Lijun Qian. Rutgers The State University of New Jersey. 94 Brett Road. Piscataway, NJ

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1 Lecture Notes of Communication Network I y: part 5 Lijun Qian Department of Electrical Engineering Rutgers The State University of New Jersey 94 Brett Road Piscataway, NJ October 6, 1998 Abstract In this lecture note, the famous Little's Theorem is briey reviewed. Little's Theorem expresses the natural idea that crowded systems are associate with long customer delays. Althrough the theorem itself is simple, it is very important due to its generality. It holds for almost every queueing system that reaches a steady state. Then the concepts of discrete-time Markov chains and continuous-time Markov chains are stated. By using the Markov chain theory as an analysis tool, the delay model of the M/M/1 queueing system is derived. The main results are highlighted. Readers can nd most of the materials in [1],[2] and [3]. ygraduate course 16:332:543 lectured by Dr. R. Yates.

2 1. Introduction This lecture note is the most basic part of delay models of data networks. In modern communication networks, the average delay todeliver a packet from the source to the destination is a very important performance measure. For real-time applications, it may be the most important one among all the Quality-of-Service(QoS) requirements. Analysis and modeling the average delay of data networks has been a hot research area for a long time. Many brilliant results were obtained[3]. But we should keep in mind that the theories are still far from reality. This lecture note is organized as follows: In section 2, Little's Theorem is briey reviewed to analysis the relation between the average delay and the average number of customers in the system when the queueing system is stable. The delay model of the M/M/1 queueing system is obtained in section 3 via the Markov chain theory. Some detailed analysis and calculations are given. 2. Little's Theorem In this section, a single queue model is considered. The model is shown in gure 1. Dene: queue server α( t) β( t)... λ µ N(t) Figure 1: A Single Queue Model N(t) : Number of customers in the system at time t. (t) : Number of customers who arrived in the interval [0;t] T i : Time spent in the system by the i th customer Then Little's theorem was stated as follows[3]: Theorem 1. Assume the queueing system reaches steady state, then N = T (1) where N is time average of number of customers in the system, is the time average arrival rate and T is the steady state time average customer delay. N, and T can be calculated as follows:

3 (t) N = lim N t ; =lim ; T=lim t!1 t!1 t P (t) i=0 T i t!1 (t) (2) Note that T i include both waiting delay W i and service delay X i. Little's Theorem is a great theoretical result due to its simple form and generality. It can be applied to almost all queueing systems as long as the system is stable, i.e., no congestions or deadlocks exist in the system. 3. Delay Model of Queueing Networks: the M/M/1 Case Queueing theory is the study of models of service systems in which tasks wait to be processed. The objectives of the theory are to predict the delays faced by tasks before their processing is completed and also the backlog of tasks waiting to be processed. A queue is a service facility equipped with a waiting room(buer). The simpliest queueing model is the M/M/1 queue. The standard denition of a A/S/n model is as follows[1]: A: arrival process. Some special cases are: M: memoryless(poisson process) D: deterministic G: general GI: independent interarrival times with given pdf S: departure process. Some special cases are: M: memoryless(service times are exponentially distributed) D: deterministic G: general GI: independent identically distributed(i.i.d.) service times n: number of servers. n = 1,2,...,m, 1. When n = 1, any customer show upwillbeserved immediately. 3.1 Discrete-time Markov Chains A brief description of the basics of discrete-time Markov chains is given in chapter 11 in [2] and at the end of chapter 3 in [3]. Let's consider discrete-time processes fx n jn =0; 1; 2;:::g where each X n is a non-negative integer random variable.

4 Denition 01. Discrete-time Markov Chain: A discrete-time Markov chain fx n jn =0; 1; 2; :::g is arandom sequence such that given X 0 ; :::; X n, the next random variable X n+1 depends only on X n through the transition probability P [X n+1 = jjx n = i; X n,1 = i n,1 ; :::; X 0 = i 0 ]=P [X n+1 = jjx n = i] =P ij (3) The value of X n summarizes all of the past history of the system needed to predict the next element X n+1 of the random sequence. We callx n the state of the system at time n. There is a xed transition probability P ij that the next state will be j given the current state is i. If weare currently in state i, at next step we must be in some state. This implies: 1X j=0 P ij =1 (4) 3.2 Continuous-time Markov Chains A continuous-time Markov chain is a process fx(t)jt 0g taking values from the set of states i =0; 1; 2;::: that has property eachtimeitenters state i 1. The time it spends in state i is exponentially distributed with rate i. P 2. When the process leaves state i, it will enter state j with probability P ij,where j P ij = 1. (It must stay at some state j.) The above denition can be found at the end of chapter 3 in [3]. 3.3 Results of M/M/1 queueing system model In this section, the statistics of the arrival and service processes of the M/M/1 system are reviewed. Then the main results are listed. Arrival Statistics: the Poisson Process. A stochastic process fa(t)jt 0g taking non-negative integer values is a Poisson Process if 1. A(t) is a counting process which represents the total number of arrivals during [0;t], where A(0) = 0. The numberofarrivals during (s; t] isa(t), A(s), where s<t. The probability mass function(pmf) is: P [A(t), A(s) =n] =f [(t,s)] n e,(t,s) n =0; 1; 2; ::: n! 0 otherwise (5)

5 2. Two non-overlapping A(t), A(s) anda(t 0 ), A(s 0 ) are independent. Then the consequences are: 1. The interarrival times are independent identically distributed(i.i.d.), and satisfy exponential distribution, i.e., the probability density function(pdf) is f (x)=f e,x x 0 0 otherwise (6) or equivalently, P [ n >xj 1 ; 2 ;:::; n,1 ]=P [A(t 1 + x), A(t 1 )=0j 1 ; 2 ; :::; n,1 ]=e,x (7) where 1 ; 2 ; :::; n are interarrival times. A(t) τ n 0 t 1 1 t +x time Figure 2: The Arrival Process of the M/M/1 system 2. P [A(t 1 + ), A(t 1 ) = 0] = e, =1, + o() (8) P [A(t 1 + ), A(t 1 )=1]=e, = + o() (9) P [A(t 1 + ), A(t 1 ) 2] = o() (10) 3. If A 1 (t);a 2 (t); :::; A k (t) are independent Poisson process with rate 1 ; 2 ;:::; k, then A(t) = A 1 (t)+a 2 (t)+::: + A k (t) isapoisson process with rate ::: + k. 4. Split a Poisson process by randomization(e.g., ip a coin) then the resulting Poisson processes are independent. Service Statistics: exponential distributed(memoryless). P [S n >x+ tjs n >t]= P [S n >x+ t] P [S n >t] = e,(x+t) e,t = e,x = P [S n >x] (11)

6 where S n are the service times. The memoryless property allows us to use the continuous-time Markov chain as an analysis tool. Denote the number of customers in the system at time t as N(t). The future numbers of customers depend on past numbers only through the present number, i.e., fn(t)jt 0g is a continuous-time Markov chain. Let's focus on the times t = ; 2; :::; k, denote N k = N(k) andp ij = P [N k+1 = jjn k = i] the transition probability. Use equation (8),(9) and (10), the following equations are easily derived P 00 = P f0 arrivals in(k; (k +1)]g =1, + o() (12) P ii = P [N k+1 = ijn k = i](i 1) (13) = P f0 arrivals 0 departures jn k = ig + P f1 arrivals 1 departures jn k = ig + ::: = P f0 departures j0 arrivals; N k = igp f0 arrivals jn k = ig + o() = e, e, + o() =1,, + o() P i;i+1 = P [N k+1 = i +1jN k = i] =P f1 arrivals 0 departures jn k = ig (14) = ( + o())(1, + o()) = + o() P i;i,1 = P [N k+1 = i, 1jN k = i] =P f0 arrivals 1 departures jn k = ig (15) = ( + o())(1, + o()) = + o() P i;j = P [N k+1 = jjn k = i](j 6= i, 1;i;i+1)=o() (16) The state transition diagram for the Markov chain fn k g is shown in gure 3 where o() is omitted. 1 λδ 1 λδ µδ 1 λδ µδ 1 λδ µδ 1 λδ µδ 1 λδ µδ λδ λδ λδ λδ n-1 n n+1 µδ µδ µδ µδ Figure 3: Discrete-time Markov Chain for the M/M/1 system Now let's consider the steady-state probabilities P j dened as P j = lim k!1 P [N k = j] (17)

7 It is obvious that P [N k+1 = j] = X i P [N k+1 = jjn k = i]p [N k = i] (18) and P j = X i P ij P i (19) Note that the probability of the system make a transition to state i + 1 when in state i is the same as the probability of the system make a transition to state i when in state i + 1. Thus the following equation holds P i P i;i+1 = P i+1 P i+1;i (20) From equation (14) and (15), we get P i = P i+1 (21) Let =, Pi+1 = Pi (22) Since P i P i = P 0 Pi i = P 0 1, =1,we gure out that P 0 =1, and P i =(1, ) i. The average number of customers in the M/M/1 system in steady-state is N = lim t!1 EfN(t)g = 1X n=0 np n = 1X n=0 n(1, ) n = (1, ) 1X n=0 1 n n,1 = (1, )( 1, )0 = 1, The relationship between the utilization factor and the average number of customers in the M/M/1 system in steady-state is shown in gure 4. (23) N Average numbers of customers Ν= ρ 1 ρ Utilization Factor 1 ρ Figure 4: Average Number of Customers in the M/M/1 System vs Utilization Factor

8 The average delay T can be calculated by Little's theorem T = N = (1, ) = 1, (24) References [1] R. Yates, Lecture Notes of Communication Networks I (16:332:543), Sep.29, [2] R. Yates and D. Goodman, Probability and Stochastic Processes, John Wiley & Sons, [3] D. Bertsekas and R. Gallager, Data Networks, 2nd. Edition, Prentice-Hall, 1992.