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This article explores the M/G/1 Markov Process, discussing the state representation, imbedded Markov Chain, transition probabilities, and the Pollaczek-Klinchin (P-K) formula. It also explores variations such as non-preemptive priority queue, queue with vacations, FDM/TDM, and polling-gated systems.
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M/G/1 [N(t), X0(t)] – state of a Markov Process N(t) - # of customers in the system X0(t) – service time already received by the customer at time t
M/G/1 • Reduced state representation • Well known X0(t) departure time imbedded Markov Chain
M/G/1 • New state number of customers left behind by a departing customer • Time between transitions b(x) – at least one customer in the system b(x) - a(x) – no customer in the system
M/G/1 nk = Pn[arrival finds k customers in the system] dk = P [departure at t leaves k customers behind] nk=dk= Pk
M/G/1 Probability that at the nth departure time the number of customers in queue was j given that at the departure time of the nth customer the number of customers in the system was i.
aj-i+1 a1 a3 a2 i-2 i-1 i i+1 i+2 j a0 M/G/1 State transition-probability diagram for the M/G/1 imbedded Markov Chain
M/G/1 • Pollaczek-Klinchin (P-K) formula
M/G/1 • P-K formula M/M/1 M/D/1
M/G/1 is the average remaining service time for the customer (if any) found in service by a new arrival (residual time)
M/G/1 M/M/1 M/D/1
Non-preemptive Priority Queue • NQk- average number in queue for priority k; • Wk - average queuing time priority k; • Rk = lk/mk- system utilization for priority k; • R - mean residual service time.
