Journal of Applied Probability

Managing queues with heterogeneous servers

Jung Hyun Kim, Hyun-Soo Ahn, and Rhonda Righter

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider several versions of the job assignment problem for an M/M/m queue with servers of different speeds. When there are two classes of customers, primary and secondary, the number of secondary customers is infinite, and idling is not permitted, we develop an intuitive proof that the optimal policy that minimizes the mean waiting time has a threshold structure. That is, for each server, there is a server-dependent threshold such that a primary customer will be assigned to that server if and only if the queue length of primary customers meets or exceeds the threshold. Our key argument can be generalized to extend the structural result to models with impatient customers, discounted waiting time, batch arrivals and services, geometrically distributed service times, and a random environment. We show how to compute the optimal thresholds, and study the impact of heterogeneity in server speeds on mean waiting times. We also apply the same machinery to the classical slow-server problem without secondary customers, and obtain more general results for the two-server case and strengthen existing results for more than two servers.

Article information

J. Appl. Probab., Volume 48, Number 2 (2011), 435-452.

First available in Project Euclid: 21 June 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx]

Multiserver queue heterogeneous server individual and social optimality


Kim, Jung Hyun; Ahn, Hyun-Soo; Righter, Rhonda. Managing queues with heterogeneous servers. J. Appl. Probab. 48 (2011), no. 2, 435--452. doi:10.1239/jap/1308662637.

Export citation


  • Agrawala, A. K., Coffman, E. G., Jr., Garey, M. R. and Tripathi, S. K. (1984). A stochastic optimization algorithm minimizing expected flow times on uniform processors. IEEE Trans. Comput. 33, 351–356.
  • Armony, M. and Mandelbaum, A. (2011). Routing and staffing in large-scale service systems: the case of homogeneous impatient customers and heterogeneous servers. Operat. Res. 59, 50–65.
  • Armony, M. and Ward, A. R. (2010). Fair dynamic routing in large-scale heterogeneous-server systems. Operat. Res. 58, 624–637.
  • De Vericourt, F. and Zhou, Y.-P. (2006). On the incomplete results for the heterogeneous server problem. Queueing Systems 52, 189–191.
  • Koole, G. (1995). A simple proof of the optimality of a threshold policy in a two-server queueing system. Systems Control Lett. 26, 301–303.
  • Kumar, P. R. and Walrand, J. (1985). Individually optimal routing in parallel systems. J. Appl. Prob. 22, 989–995.
  • Lin, W. and Kumar, P. R. (1984). Optimal control of a queueing system with two heterogeneous servers. IEEE Trans. Automatic Control 29, 696–703.
  • Luh, H. P. and Viniotis, I. (2002). Threshold control policies for heterogeneous server systems. Math. Meth. Operat. Res. 55, 121–142.
  • Righter, R. (1988). Job scheduling to minimize expected weighted flowtime on uniform processors. Systems Control Lett. 10, 211–216.
  • Righter, R. and Xu, S. (1991). Scheduling jobs on heterogeneous processors. Ann. Operat. Res. 29, 587–601.
  • Righter, R. and Xu, S. H. (1991). Scheduling jobs on nonidentical IFR processors to minimize general cost functions. Adv. Appl. Prob. 23, 909–924.
  • Rosberg, Z. and Makowski, A. M. (1990). Optimal routing to parallel heterogeneous servers–-small arrival rates. IEEE Trans. Automatic Control 35, 789–796.
  • Rykov, V. V. (2001). Monotone control of queueing systems with heterogeneous servers. Queueing Systems 37, 391–403.
  • Singh, V. P. and Prasad, J. (1976). A heterogeneous system with finite waiting space. J. Eng. Math. 10, 125–134.
  • Stockbridge, R. H. (1991). A martingale approach to the slow server problem. J. Appl. Prob. 28, 480–486.
  • Weber, R. (1993). On a conjecture about assigning jobs to processors of differing speeds. IEEE Trans. Automatic Control 38, 166–170.
  • Walrand, J. (1984). A note on: “Optimal control of a queuing system with two heterogeneous servers”. Systems Control Lett. 4, 131–134.
  • Xu, S. H. (1994). A duality approach to admission and scheduling controls of queues. Queueing Systems 18, 273–300.
  • Xu, S. H. and Shanthikumar, J. G. (1993). Optimal expulsion control–-a dual approach to admission control of an ordered-entry system. Operat. Res. 41, 1137–1152.