Some Finite Waiting Space Bulk Queueing Systems

Summary

The imbedded Markov chain {Q_n} of the single server M^X/G^Y/1 bulk queue, with Poisson input and continuously busy server, satisfies the recurrence relation given by Bhat [1964]

$$ {Q_{n + 1}} = {\left( {{Q_n} - {Y_n}} \right)^ + } + {X_{n + 1}} $$

(1)

where Q_n is the number of customers in the system at epoch (t_n+0) , immediately after the end of the n^th service period, Y_n is the number of customers which can be served by the server in the service period commencing at (t_n+0), and X_n+1 is the number of customers arriving in (t_n, t_n+1]. Finally, (.)⁺ is defined by Z⁺ = max(Z,0). If {X_n} and {Y_n} are sequences of independently distributed random variables, then the sequence {Q_n} will be a Markov chain.

For a finite waiting space M^X/G^Y/1,K queue, in which the number of customers in the system must not exceed K at any time, equation (1) for the continuously busy server must be replaced by

$$ {Q_{n + 1}} = K + {[{\left( {{Q_n} - {Y_n}} \right)^ + } + {X_{n + 1}} - K]^ - } $$

(2)

where (.)⁻ is defined by Z⁻ = min(Z,0).

Equation (2) is used, following Cohen [1969], to obtain and solve an integral equation (of the type used by Pollaczek [1957]) for the time-dependent state probabilities P[Q_n = j|Q₁ = z] and for the long-run state probabilities P[Q = j], for j = 0,1,...,K. The formulas obtained are not simple.

The time-dependent and long-run state probabilities for the M/G/1,K queue given by Cohen [1969], can be obtained by specialization from our results for the M^X/G^Y/1,K queue.