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# Queueing Networks

Pollaczek-Khintchine Formulas This section deals with formulas of an M/G/1 queueing system at which the customers arrive according to a Poisson-process with parameter λ, the service time is arbitrarily distributed, there is no restriction to the number of customers staying in the system, and they are serviced according to the order of their arrivals, that is the service discipline is FCFS. These formulas are treated almost every book on queueing theory but the notation is quite different. Each author prefers his own designation and as a consequence it is very difficult to find the proper form. We make difference between the type of formulas, the mean value and transform ones. Of course, the first ones are much easier to obtain. Independently of each other, Pollaczek and Khintchine derived them in the period 1930-50. Felix Pollaczek, 1892–1981 Alexander Y. Khintchine, 1894–1959 As usual we need some notations. In steady state let us denote by • N number of customers in the system, • P(z) the generating function of N, that is P(z) = E(z N ), • B∗ (s) = R ∞ 0 e −stdB(t) the Laplace-Stieltjes transform of the service time S, • W∗ (s) the Laplace-Stieltjes transform of the waiting time in the system, or response time T , • W∗ q (s) the Laplace-Stieltjes transform of the waiting time in the queue Tq, • C 2 S = V ar(S) E2(S) squared coefficient of variation of service time S, • L = E(N), Lq = E(Nq) average number of customers in the system, queue, respectively, Queueing Theory and its Applications, A Personal View 21 • W = E(T ), Wq = E(Tq) mean waiting time in the system, in the queue, respectively, • ρ = λE(S). Hence, the mean value formulas are as follows: L = ρ + ρ 2 1 + C 2 S 2(1 − ρ) , Lq = ρ 2 1 + C 2 S 2(1 − ρ) , or by using the Little’s law we have W = E(S)(1 + ρ 1 + C 2 S 2(1 − ρ) ), Wq = E(S)ρ 1 + C 2 S 2(1 − ρ) . The transform formulas or equations can be written as P(z) = (1 − ρ)(z − 1)B∗ (λ − λz) z − B∗(λ − λz) , P(z) = W∗ (λ − λz), W∗ (s) = (1 − ρ)sB∗ (s) s − λ(1 − B∗(s)), W∗ q (s) = (1 − ρ)s s − λ(1 − B∗(s)).