The Workload in the M/G/1 Queue with Work Removal

Abstract

We consider an MIG/ I queue with the special feature of additional negative customers, who arrive according to a Poisson process. Negative customers require no service, but at their arrival a stochastic amount of work is instantaneously removed from the system. We show that the workload distribution in this MIG/ 1 queue with negative customers equals the waiting time distribution in a GI/G/l queue with ordinary customers only; the effect of the negative customers is incorporated in the new arrival process.

A Note on Negative Customers, GI/G/1 Workload, and Risk Processes

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This note illustrates that a combination of the approach in our previous papers (Boucherie and Boxma, 1996, Probability in the Engineering and Informational Sciences 10: 261-277; Jain and Sigman, 1996, Probability in the Engineering and Informational Sciences I 0: 519-531) directly leads to a Pollaczek-Khintchine form for the workload in a queue with negative customers. The same technique is also applied to risk processes with lump additions.

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“…Consider the M/G/1 queue with additional removal of work as studied by Boucherie and Boxma [2] and Jain and Sigman [8]. Customers arrive according to a Poisson process with rate f..+.…”

“…(l) suggests the following transformation, relating the workload at arrival epochs of positive customers to this workload at arrival epochs in a standard GI/G/l queue [2] (cf. )=I (2) Clearly, Eq. (1) is the recurrence relation for the waiting time in a Gl/G/1 queue with service times Bn but extended interarriva/ times 1;.…”

“…The transformation idea of [2], lengthening of interarrival times to take work removal into account, gives direct access to the literature on the Gl/G/1 queue; see, for example, Cohen [3], in particular for the useful detailed results for the case that either the interarrival time distribution or the service time distribution has a rational LST.…”

A Blood Bank Model with Perishable Blood and Demand Impatience

“…In addition, the steady-state amounts of demand and of blood have an atom at 0 (since ξ d and ξ b are no longer zero, the demand and blood processes can reach 0). Interestingly, the model of this section is closely related to the model with workload removal that is considered in Boucherie and Boxma (1996). There an M/G/1 queue is studied with the extra feature that, at Poisson epochs, a stochastic amount of work is removed.…”