M/G/1

1. M/G/1 Cheng-Fu Chou

2. Residual Life • Inter-arrival time of bus is exponential w/ rate l while hippie arrives at an arbitrary instant in time • Question: How long must the hippie wait, on the average , till the bus comes along? • Answer 1: Because the average inter-arrival time is 1/l, therefore 1/2l

3. Residual Life (cont.) • Answer 2: because of memoryless, it has to wait 1/l • General Result

4. Derivation

6. M/G/1 • M/G/1 • a(t) = le –lt • b(t) = general • Describe the state [N(t), X0(t)] • N(t): the no. of customers present at time t • X0(t): service time already received by the customer in service at time t • Rather than using this approach, we use “the method of the imbedded Markov Chain”

7. Imbedded Markov Chain [N(t), X0(t)] • Select the “departure” points, we therefore eliminate X0(t) • Now N(t) is the no. of customer left behind by a departure customer. (HW) • For Poisson arrival pk(t) = rk(t) • If in any system (even in non-Markovian) where N(t) makes discontinuous changes in size (plus or minus) one, then • rk = dk = prob[departure leaves k customers behind] • Therefore, for M/G/1 • rk = dk = pk

10. Mean Queue Length

17. This is the famous Pollaczek – Khinchin Mean Value Formula

18. Examples

19. Mean Residual Service Time • Wi: waiting time in queue of the i-th customer • Ri: residual service time seen by the i-th customer • Xi: service time of the i-th customer • Ni: # of customers found waiting in the queue by the i-th customer upon arrival

21. Residual Service Time rt x1 time t x2 xM(t) x1

23. Distribution of Number in the System

25. Ex • Q(z) for M/M/1

26. Sol.

27. Waiting Time Distribution

28. Ex • Response time distribution for M/M/1 • Waiting time distribution for M/M/1

29. Response Time Distribution

30. Waiting Time Distribution