Analysis of M/M/1, M/M/c, and M/M/c/c

1. Analysis of M/M/1, M/M/c, and M/M/c/c

2. AGENDA • Markov Chain • Basic Queueing • Analysis of M/M/1 • Analysis of M/M/c • Analysis of M/M/c/c

3. Markov Chain • Markov Chain is a random process with the property that the next state depends only on the current state • Transition probabilities Pij • Transition probability matrix P=[Pij]

4. Example of Markov Chain • State: • If the weather in Hawaii today is known to be sunny, it can be represented by a vector x(0) = [1 0] • That is “sunny” entry is 100% and “rainy” entry is 0% • Probability Model: • Let the weather model be represented by a transition matrix 0.1 0.9 0.5 0.5

5. What do we know about tomorrow’s weather? • The weather on day 1 can be predicted by Interpretation?? There is a 90% chance that tomorrow will also be sunny 90% 10%

6. What about…. • Can also repeat the same procedure: ? 86% 14%

7. In general, what’s the weather like in Hawaii? • We need to find the “Steady State Probability vector” • That is  is unchanged by P 83.3% 16.7%

8. Buffer Server(s) Departures Arrivals Queued In Service Basic Queueing Model • A queue models any service station with: • One or multiple servers • A waiting area or buffer • Customers arrive to receive service • A customer that upon arrival does not find a free server is waits in the buffer

9. Characteristics of a Queue • Number of servers m: one, multiple, infinite • Buffer size b • Service discipline (scheduling): FCFS, LCFS, Processor Sharing (PS), etc • Arrival process • Service statistics m b

10. Arrival Process • : interarrival time between customers n and n+1 • is a random variable • is a stochastic process • Interarrival times are identically distributed and have a common mean • l is called the arrival rate

11. Service-Time Process • : service time of customer n at the server • is a stochastic process • Service times are identically distributed with common mean • m is called the service rate • For packets, are the service times really random?

12. Queue Descriptors • Generic descriptor: A/B/m/k • A denotes the arrival process • For Poisson arrivals we use M (for Markovian) • B denotes the service-time distribution • M: exponential distribution • D: deterministic service times • G: general distribution • m is the number of servers • k is the max number of customers allowed in the system – either in the buffer or in service • k is omitted when the buffer size is infinite

13. Queue Descriptors: Examples • M/M/1: Poisson arrivals, exponentially distributed service times, one server, infinite buffer • M/M/m: same as previous with m servers • M/M/m/m: Poisson arrivals, exponentially distributed service times, m server, no buffering • M/G/1: Poisson arrivals, identically distributed service times follows a general distribution, one server, infinite buffer • */D/∞ : A constant delay system

14. M/M/1 Queue • Customers arrive according to a Poisson of rate so the interarrival times are iid exponential RV’s with mean 1/  • Service times are iid exponential RV’s with mean 1/ and are independent of the interarrival time • Single-Server system • Queue size is infinite

15. Required Concept: Global Balance Equations • Markov chain with infinite number of states • Global Balance Equations (GBE) • is the frequency of transitions from j to i

16.    . . . 0 2 1 j j+1    Analysis of M/M/1 Global Balance equation for steady-state probabilities are P0 = P1(1) ( + )Pj= Pj -1 +  Pj+1 ;j = 1,2,3… (2) A steady state solution exists when  = < 1 Rewriting second part of (2) Pj -  Pj+1 = Pj-1 - Pj;j = 1,2,3… (3)

17. Analysis of M/M/1 (cont) From (1) it is easy to see that P1 - P2 = 0 (4) Thus (3) becomes Pj-1 =  Pj Pj =  Pj-1 ;j = 1,2,3… By induction Pj =  j P0 We obtain P0 by noting that sum of probabilities = 1 where the series converges if and only if  <1 Pj = (1 – ) j ;j = 1,2,3… (5)

18. Analysis of M/M/1 (cont1) Note: The condition must be met if the system is to be stable. That is N(t) does not grow. The mean number of customers in the system is given by Using (6) together with Little’s formula, the mean delay in the system is

19. Analysis of M/M/1 (cont1) The mean waiting time in queue is given by By using Little’s formula, the mean number of customers in the queue is The server utilization is given by

20. E[N]  0.5 0.5 1 1 M/M/1: Performance Metrics 

21.       . . . . . . 1 0 2 C-2 C-1 C C+1 j j+1  2 (c-1) c c c Multi-Server Systems: M/M/c • Arrival occurs at a rate of  • Departure rate is k when k servers are busy • When k servers are busy, then the time until the next departure is given by X = min(1, 2,…, k), (1) where i are iid exponential RV’s with parameter . Then, P[X > t] = P[min (1, 2,…, k) > t] = P[1>t] P[2>t] … P[k>t] = e-kt (2) • Thus the average time until the next departure is 1/k

22. Analysis of M/M/c From the general solution for birth-and-death processes, the probabilities of the first c states are obtained from the following recursion: The probabilities for states equal to or greater than c are obtained from the following recursion:

23. Analysis of M/M/c (cont1) Finally, P0is obtained from the normalization condition: The system is stable if  < 1 or  < c (6) The final form of P0 is The probability that an arriving customer finds all servers busy is

24. Analysis of M/M/c (cont2) The probability in (8) is called ERLANG C formula and is denoted by C(c,a): The mean number of customers in the queue is given by: The mean waiting time is found from Little’s formula:

25. Analysis of M/M/c (cont3) The mean total time in the system is E[T] = E[W] + E[] = E[W] + 1/ (12) Finally, the mean number in the system is found from Little’s formula: E[N] = E[T] = E[Nq] +a (13)

26.     . . . 1 0 2 C-2 C-1 C  2 (c-1) c Homework : Analysis of M/M/c/c • What are the systems that use M/M/c/c? • Verify for yourself that: • The probability of blocking or Pc is called the Erlang B formula