MG1 ChengFu Chou CMLab CSIE NTU Residual Life

M/G/1 Cheng-Fu Chou, CMLab, CSIE, NTU

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

Residual Life (cont. ) q Answer 2: because of memoryless, it has to wait 1/l q General Result P. Cheng-Fu Chou, CMLAB, CSIE, NTU

Derivation P. Cheng-Fu Chou, CMLAB, CSIE, NTU

P. Cheng-Fu Chou, CMLAB, CSIE, NTU

M/G/1 q M/G/1 – a(t) = le –lt – b(t) = general q Describe the state [N(t), X 0(t)] – N(t): the no. of customers present at time t – X 0(t): service time already received by the customer in service at time t q Rather than using this approach, we use “the method of the imbedded Markov Chain” P. Cheng-Fu Chou, CMLAB, CSIE, NTU

Imbedded Markov Chain [N(t), X 0(t)] q Select the “departure” points, we therefore eliminate X 0(t) q 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 o rk = dk = prob[departure leaves k customers behind] – Therefore, for M/G/1 o rk = dk = p k P. Cheng-Fu Chou, CMLAB, CSIE, NTU

P. Cheng-Fu Chou, CMLAB, CSIE, NTU

Mean Queue Length P. Cheng-Fu Chou, CMLAB, CSIE, NTU

P. Cheng-Fu Chou, CMLAB, CSIE, NTU

q This is the famous Pollaczek – Khinchin Mean Value Formula P. Cheng-Fu Chou, CMLAB, CSIE, NTU

Examples P. Cheng-Fu Chou, CMLAB, CSIE, NTU

Mean Residual Service Time q q 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 – P. Cheng-Fu Chou, CMLAB, CSIE, NTU