Analysis of a dynamic assignment of impatient customers to parallel queues

Analysis of a dynamic assignment of impatient customers to parallel queues

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Article ID: iaor20115201
Volume: 67
Issue: 3
Start Page Number: 251
End Page Number: 273
Publication Date: Mar 2011
Journal: Queueing Systems
Authors:
Keywords: parallel queues, Poisson process
Abstract:

Consider a number of parallel queues, each with an arbitrary capacity and multiple identical exponential servers. The service discipline in each queue is first‐come‐first‐served (FCFS). Customers arrive according to a state‐dependent Poisson process. Upon arrival, a customer joins a queue according to a state‐dependent policy or leaves the system immediately if it is full. No jockeying among queues is allowed. An incoming customer to a parallel queue has a general patience time dependent on that queue after which he/she must depart from the system immediately. Parallel queues are of two types: type 1, wherein the impatience mechanism acts on the waiting time; or type 2, a single server queue wherein the impatience acts on the sojourn time. We prove a key result, namely, that the state process of the system in the long run converges in distribution to a well‐defined Markov process. Closed‐form solutions for the probability density function of the virtual waiting time of a queue of type 1 or the offered sojourn time of a queue of type 2 in a given state are derived which are, interestingly, found to depend only on the local state of the queue. The efficacy of the approach is illustrated by some numerical examples.

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