Skip to main content

Scheduling networks of queues: Heavy traffic analysis of a simple open network


We consider a queueing network with two single-server stations and two types of customers. Customers of type A require service only at station 1 and customers of type B require service first at station 1 and then at station 2. Each server has a different general service time distribution, and each customer type has a different general interarrival time distribution. The problem is to find a dynamic sequencing policy at station 1 that minimizes the long-run average expected number of customers in the system.

The scheduling problem is approximated by a dynamic control problem involving Brownian motion. A reformulation of this control problem is solved, and the solution is interpreted in terms of the queueing system in order to obtain an effective sequencing policy. Also, a pathwise lower bound (for any sequencing policy) is obtained for the total number of customers in the network. We show via simulation that the relative difference between the performance of the proposed policy and the pathwise lower bound becomes small as the load on the network is increased toward the heavy traffic limit.

This is a preview of subscription content, access via your institution.


  1. J.M. Harrison,Brownian Motion and Stochastic Flow Systems (John Wiley and Sons, New York, 1985).

    Google Scholar 

  2. J.M. Harrison, Brownian models of queueing networks with heterogeneous customer populations, In:Stochastic Differential Systems, Stochastic Control Theory and Applications, eds. W. Fleming and P.L. Lions, IMA Vol. 10 (Springer-Verlag, New York, 1988) 147–186.

    Google Scholar 

  3. J.M. Harrison and L.M. Wein, Scheduling networks of queues: heavy traffic analysis of a two-station closed network, Operations Research (1989) to appear.

  4. F.P. Kelly,Reversibility and Stochastic Networks (John Wiley and Sons, New York, 1979).

    Google Scholar 

  5. C.N. Laws and G.M. Louth, Dynamic scheduling of a four station network, Probability in the Engr. and Inf. Sciences (1988) submitted.

  6. M. Moustafa, Optimal scheduling in networks of queues, unpublished Ph. D. disseration, Program in Operations Research, North Carolina State University, Raleigh, NC, 1987.

    Google Scholar 

  7. L.M. Wein, Optimal control of a two-station Brownian network, Math. of Operations Research (1988) to appear.

  8. L.M. Wein, Scheduling networks of queues: heavy traffic analysis of a two-station network with controllable inputs, Operations Research (1989) to appear.

  9. P. Yang, Pathwise solutions for a class of linear stochastic systems, unpublished Ph.D. thesis, Dept. of Operations Research, Stanford University, Stanford, CA, 1988.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and Permissions

About this article

Cite this article

Harrison, J.M., Wein, L.M. Scheduling networks of queues: Heavy traffic analysis of a simple open network. Queueing Syst 5, 265–279 (1989).

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI:


  • Brownian approximations
  • networks of queues
  • scheduling theory
  • heavy traffic analysis
  • dynamic priority
  • pathwise solution