Abstract
We consider Kelly networks with shuffling of customers within each queue. Specifically, each arrival, departure or movement of a customer from one queue to another triggers a shuffle of the other customers at each queue. The shuffle distribution may depend on the network state and on the customer that triggers the shuffle. We prove that the stationary distribution of the network state remains the same as without shuffling. In particular, Kelly networks with shuffling have the product form. Moreover, the insensitivity property is preserved for symmetric queues.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baskett, F., Chandy, K.M., Muntz, R.R., Palacios, F.G.: Open, closed and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260 (1975)
Bertsekas, D., Gallager, R.: Data Networks. Prentice-Hall, Englewood Cliffs (1987)
Boucherie, R., van Dijk, N.: Product forms for queueing networks with state dependent multiple job transitions. Adv. Appl. Probab. 23, 152–187 (1991)
Chao, X., Miyazawa, M., Pinedo, M.: Queueing Networks, Customers, Signals and Product Form Solutions. Wiley, New York (1999)
Daduna, H.: Stochastic networks with product form equilibrium. In: Shanbhag, D.N., Rao, C.R. (eds.) Stochastic Processes: Theory and Methods, pp. 309–364. Elsevier, Amsterdam (2001)
Daduna, H.: Queueing Networks with Discrete Time Scale: Explicit Expressions for the Steady State Behavior of Discrete Time Stochastic Networks. Lecture Notes in Computer Science, vol. 2046. Springer, Berlin (2001)
Daduna, H., Schassberger, R.: Networks of queues in discrete time. Z. Oper. Res. 27, 159–175 (1983)
Fu, M.C., Govil, M.K.: Queueing theory in manufacturing: A review. J. Manuf. Syst. 18, 214–240 (1999)
Hordijk, A., Schassberger, S.: Weak convergence of generalized semi-Markov processes. Stoch. Process. Appl. 12, 271–291 (1982)
Hordijk, A., van Dijk, N.: Adjoint processes, job local balance and insensitivity for stochastic networks. Bull: 44 Session Int. Stat. Inst. 50, 776–788 (1982)
Jackson, J.R.: Networks of waiting lines. Oper. Res. 5, 518–521 (1957)
Kelly, F.P.: Reversibility and Stochastic Networks. Wiley, New York (1979)
Kleinrock, L.: Queueing Systems, vol. 2. Wiley, New York (1975)
Miyazawa, M.: Insensitivity and product-form decomposability of reallocatable GSMP. Adv. Appl. Probab. 25, 415–437 (1993)
Miyazawa, M., Schassberger, S., Schmidt, S.: On the structure of an insensitive generalized semi-Markov process with reallocation and point-process input. Adv. Appl. Probab. 27, 203–225 (1995)
Schassberger, R.: The insensitivity of stationary distributions in networks of queues. Adv. Appl. Probab. 10, 906–912 (1978)
Serfozo, R.F.: Introduction to Stochastic Networks. Springer, Berlin (1999)
Yashkov, S.F.: Properties of invariance of probabilistic models of adaptive scheduling in shared use systems. Autom. Control Comput. Sci. 14, 46–51 (1980)
Yates, R.D.: High speed round-robin queueing networks. Ph.D. Thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (1990)
Yates, R.D.: Analysis of discrete time queues via the reversed process. Queueing Syst. 18, 107–116 (1994)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bonald, T., Tran, MA. On Kelly networks with shuffling. Queueing Syst 59, 53–61 (2008). https://doi.org/10.1007/s11134-008-9075-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-008-9075-8