The Busy Period of an M/G/1 Queue with Customer Impatience

Abstract

We consider an M/G/1 queue in which an arriving customer doesn't enter the system whenever its virtual waiting time, i.e., the amount of work seen upon arrival, is larger than a certain random patience time. We determine the busy period distribution for various choices of the patience time distribution. The main cases under consideration are exponential patience and a discrete patience distribution.

A New Look at Organ Transplantation Models and Double Matching Queues

“…From the above description the OFF period has the same law as the busy period in the M/M/1 + G queue with arrival rate μ, service rate λ, and patience distribution H(·). It should be noted that the busy period distribution for M/M/1 + G is only known in special cases-in particular for discrete patience times (see [4]; see [3] for preliminary results on the busy period for M/G/1 + G). Letf (·) be the steady-state density of the workload of this queue.…”

“…The renewal reward theorem now shows that the long-run fraction of time that the shelf is empty is equal to σ = f (1)/f (0), where f (1) andf (0) =k are given by (4) and (21), respectively.…”

“…From our assumptions, the strong Markov property, and the lack-of-memory property of the jump sizes, it follows that L has the same law as the busy period of the M/M/1 + M queue with arrival rate μ, service rate λ, and patience rate η. For its LST (β), no analytic expression is known; however, in [4,Sect. 3] it is given in the form of a continued fraction that can be derived from the recursion (β) = 1 (β) and…”

The M/G/1+G queue revisited

“…In this section we present a novel, algorithmic, method for obtaining the LST of B (see [7,15] for other studies of the busy period in M/G/1 + G w ) and some more general level crossing results. Let {V (t) : t ≥ 0} be the virtual waiting-time process and define the stopping time…”

“…For other variants of M/G/1 models with restricted accessibility, see [6,10]. In a recent study [7], the busy period distribution in M/G/1 + G w is obtained for various choices of the patience time distribution. The main cases under consideration are exponential patience and a discrete patience distribution.…”