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A queueing network with a single cyclically roving server

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Abstract

A queueingnetwork that is served by asingle server in a cyclic order is analyzed in this paper. Customers arrive at the queues from outside the network according to independent Poisson processes. Upon completion of his service, a customer mayleave the network, berouted to another queue in the network orrejoin the same queue for another portion of service. The single server moves through the different queues of the network in a cyclic manner. Whenever the server arrives at a queue (polls the queue), he serves the waiting customers in that queue according to some service discipline. Both the gated and the exhaustive disciplines are considered. When moving from one queue to the next queue, the server incurs a switch-over period. This queueing network model has many applications in communication, computer, robotics and manufacturing systems. Examples include token rings, single-processor multi-task systems and others. For this model, we derive the generating function and the expected number of customers present in the network queues at arbitrary epochs, and compute the expected values of the delays observed by the customers. In addition, we derive the expected delay of customers that follow a specific route in the network, and we introduce pseudo-conservation laws for this network of queues.

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Abbreviations

Bi, B *i (s):

service time of a customer at queue i and its LST

bi, bi (2) :

mean and second moment of Bi

Ri, R *i (s):

duration of switch-over period from queue i and its LST

ri, ri :

mean and second moment of Ri

r, r(2) :

mean and second moment of σ Ni =1Ri

λi :

external arrival rate of type-i customers

γi :

total arrival rate into queue i

ρi :

utilization of queue i; ρi=γi

ρ:

system utilization σ Ni =1ρi

c=E[C]:

the expected cycle length

X ji :

number of customers in queue j when queue i is polled

Xi=X ii :

number of customers residing in queue i when it is polled

fi(j):

\(\left\{ {\begin{array}{*{20}c} {E(X_i^j X_i^k ), j \ne k,} \\ {E[X_i^j )^2 ] - E(X_i^j ), j = k;} \\ \end{array} } \right.\)

X *i :

number of customers residing in queue i at an arbitrary moment

Yi :

the duration of a service period of queue i

Wi,Ti :

the waiting time and sojourn time of an arbitary customer at queue i

F*(z1, z2,..., zN):

GF of number of customers present at the queues at arbitrary moments

Fi(z1, z2,..., zN):

GF of number of customers present at the queues at polling instants of queue i

¯Fi(z1, z2,...,zN):

GF of number of customers present at the queues at switching instants of queue i

Vi(z1, z2,..., zN):

GF of number of customers present at the queues at service initiation instants at queue i

¯Vi(z1,z2,...,zN):

GF of number of customers present at the queues at service completion instants at queue i

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Authors and Affiliations

  1. Electrical Engineering Department, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Moshe Sidi

  2. Computer Science Department, Tel-Aviv University, 69978, Tel-Aviv, Israel

    Hanoch Levy

  3. IBM Research Division, Zurich Research Laboratory, Säumerstrasse 4, CH-8803, Rüschlikon, Switzerland

    Steve W. Fuhrmann

Authors
  1. Moshe Sidi
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  2. Hanoch Levy
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  3. Steve W. Fuhrmann
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Additional information

The work of this author was supported by the Bernstein Fund for the Promotion of Research and by the Fund for the Promotion of Research at the Technion.

Part of this work was done while H. Levy was with AT&T Bell Laboratories.

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Sidi, M., Levy, H. & Fuhrmann, S.W. A queueing network with a single cyclically roving server. Queueing Syst 11, 121–144 (1992). https://doi.org/10.1007/BF01159291

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  • DOI: https://doi.org/10.1007/BF01159291

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