Convexity Properties and Comparative Statics for M/M/S Queues with Balking and Reneging
31 Pages Posted: 31 Oct 2008
Date Written: 2005
We use sample path arguments to derive convexity properties of an M/M/S queue withimpatient customers that balk and renege. First, assuming that the balking probability andreneging 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 salesrate is convex decreasing in the capacity. Finally, we employ a sample-path sub-modularityapproach to comparative statics. That is, we employ sample path arguments to show how theoptimal 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 thebalking probability, but is not monotone in the reneging rate. This means, surprisingly, thatfailure 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 duringservice, produces qualitatively different convexity properties and comparative statics.
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