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Convexity Properties and Comparative Statics for M/M/S Queues with Balking and Reneging

Authors: Armony, Mor
Plambeck, Erica
Seshadri, Sridhar
Issue Date: 2005
Publisher: Stern School of Business, New York University
Series/Report no.: OM-2005-09
Abstract: We use sample path arguments to derive convexity properties of an M/M/S queue with impatient customers that balk and renege. First, assuming that the balking probability and reneging rate are increasing and concave in the total number of customers in the system (head-count), we prove that the expected head-count is convex decreasing in the capacity (service rate). Second, with linear reneging and balking, we show that the expected lost sales rate is convex decreasing in the capacity. Finally, we employ a sample-path sub-modularity approach to comparative statics. That is, we employ sample path arguments to show how the optimal capacity changes as we vary the parameters of customer demand and impatience. We find that the optimal capacity increases in the demand rate and decreases with the balking probability, but is not monotone in the reneging rate. This means, surprisingly, that failure to account for customersâ reneging may result in over-investment in capacity. Finally, we show that a seemingly minor change in system structure, customer commitment during service, produces qualitatively different convexity properties and comparative statics.
Appears in Collections:IOMS: Operations Management Working Papers

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