Erlang loss systems with shortest idle server first service discipline: Maintenance considerations

Mahboubeh Madadi, Mohammadhossein Heydari, Lisa Maillart, Richard Cassady and Shengfan Zhang

IISE Transactions, 2023, vol. 55, issue 10, 1008-1021

Abstract: We consider a variation of an Erlang loss system in which jobs are routed to servers according to the Shortest Idle Server First service discipline. Specifically, we consider a system in which idle servers are arranged in a stack; servers are returned to the top of the stack upon service completion; and arriving jobs are assigned to the server currently at the top of the stack. When busy, servers accumulate age and incur an age-dependent operating cost. For such systems, we (i) formulate a continuous-time Markov chain model to characterize the system’s transient behavior, and (ii) develop maintenance policies consisting of two possible actions: server group replacement and stack inversion. The stack inversion may be performed at any time prior to group replacement to achieve a more evenly distributed utilization among servers. We develop an optimization model to determine the optimal inversion and replacement times so as to minimize the long-run expected cost rate. Because the model is nonlinear and non-convex, we develop a set of algorithms to solve for the optimal replacement and inversion time. Lastly, we establish a lower bound for the inversion cost threshold below which it is optimal to invert the stack of servers before their replacement.

Date: 2023
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DOI: 10.1080/24725854.2022.2149906

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