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

31 Pages Posted: 31 Oct 2008

See all articles by Mor Armony

Mor Armony

New York University (NYU) - Department of Information, Operations, and Management Sciences

Erica L. Plambeck

Stanford Graduate School of Business

Sridhar Seshadri

University of Illinois at Urbana Champaign; Indian School of Business

Date Written: 2005

Abstract

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.

Suggested Citation

Armony, Mor and Plambeck, Erica L. and Seshadri, Sridhar, Convexity Properties and Comparative Statics for M/M/S Queues with Balking and Reneging (2005). Operations Management Working Papers Series, Vol. , pp. -, 2005. Available at SSRN: https://ssrn.com/abstract=1290212

Mor Armony (Contact Author)

New York University (NYU) - Department of Information, Operations, and Management Sciences ( email )

44 West Fourth Street
New York, NY 10012
United States
(212) 998-0291 (Phone)
(212) 995-4227 (Fax)

Erica L. Plambeck

Stanford Graduate School of Business ( email )

655 Knight Way
Stanford, CA 94305-5015
United States

Sridhar Seshadri

University of Illinois at Urbana Champaign ( email )

1206 South Sixth Street
Champaign, IL 61820
United States

Indian School of Business ( email )

Hyderabad, Gachibowli 500 019
India

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