An N Server Cutoff Priority Queue Where Arriving Customers Request a Random Number of Servers

We consider a multi-priority, N-server, Poisson arrival, nonpreemptive queue, motivated by police applications. The number of servers requested by an arrival has a known priority dependent probability distribution. All servers requested by a customer must start service simultaneously; the servers' service times are independent and exponentially distributed with parameter μ, independent of priority, server identity or system state. In order to save available servers for higher priority customers, arriving customers of each lower priority are deliberately queued whenever the number of servers busy equals or exceeds a given priority-dependent cutoff number. Whenever all higher priority queues are empty, the longest waiting priority i customer will enter service the instant there is a service completion from a state having precisely N_i − k + 1 servers busy, where k is the number of servers requested by the customer and N_i is the server cutoff number for priority i. The queueing discipline is in a sense HOL by priorities. We derive the priority i waiting time distribution (in transform domain) and other system statistics. Illustrative computational results are given.