## Annals of Probability

### The existence of fixed points for the $\cdot/GI/1$ queue

#### Abstract

A celebrated theorem of Burke's asserts that the Poisson process is a fixed point for a stable exponential single server queue; that is, when the arrival process is Poisson, the equilibrium departure process is Poisson of the same rate. This paper considers the following question: Do fixed points exist for queues which dispense i.i.d. services of finite mean, but otherwise of arbitrary distribution (i.e., the so-called $\cdot/\mathit{GI}/1/\infty/\mathit{FCFS}$ queues)? We show that if the service time $S$ is nonconstant and satisfies \mbox{$\int\!\! P\{S \geq u\}^{1/2}\,du <\infty$}, then there is an unbounded set $\fF\subset(E[S],\infty)$ such that for each $\alpha\in\fF$ there exists a unique ergodic fixed point with mean inter-arrival time equal to $\alpha$. We conjecture that in fact $\fF=(E[S],\infty)$.

#### Article information

Source
Ann. Probab., Volume 31, Number 4 (2003), 2216-2236.

Dates
First available in Project Euclid: 12 November 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1068646383

Digital Object Identifier
doi:10.1214/aop/1068646383

Mathematical Reviews number (MathSciNet)
MR2016617

Zentralblatt MATH identifier
1084.60057

#### Citation

Mairesse, Jean; Prabhakar, Balaji. The existence of fixed points for the $\cdot/GI/1$ queue. Ann. Probab. 31 (2003), no. 4, 2216--2236. doi:10.1214/aop/1068646383. https://projecteuclid.org/euclid.aop/1068646383

#### References

• Anantharam, V. (1993). Uniqueness of stationary ergodic fixed point for a $\cdot/M/K$ node. Ann. Appl. Probab. 3 154--172. [Correction (1994) Ann. Appl. Probab. 4 607.]
• Baccelli, F., Borovkov, A. and Mairesse, J. (2000). Asymptotic results on infinite tandem queueing networks. Probab. Theory Related Fields 118 365--405.
• Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• Borovkov, A. (1976). Stochastic Processes in Queueing Theory. Springer, Berlin. [Russian edition (1972), Nauka, Moscow.]
• Borovkov, A. (1984). Asymptotic Methods in Queueing Theory. Wiley, New York. [Russian edition (1980), Nauka, Moscow.]
• Brandt, A., Franken, P. and Lisek, B. (1990). Stationary Stochastic Models. Wiley, New York.
• Burke, P. (1956). The output of a queueing system. Oper. Res. 4 699--704.
• Chang, C. S. (1994). On the input-output map of a $G/G/1$ queue. J. Appl. Probab. 31 1128--1133.
• Daley, D. and Rolski, T. (1992). Finiteness of waiting-time moments in general stationary single-server queues. Ann. Appl. Probab. 2 987--1008.
• Dudley, R. (1989). Real Analysis and Probability. Wadsworth & Brooks/Cole, Belmont, CA.
• Ethier, S. and Kurtz, T. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
• Glynn, P. and Whitt, W. (1991). Departures from many queues in series. Ann. Appl. Probab. 1 546--572.
• Gray, R. (1988). Probability, Random Processes, and Ergodic Properties. Springer, Berlin.
• Kamae, T., Krengel, U. and O'Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Probab. 5 899--912.
• Loynes, R. (1962). The stability of a queue with non-independent interarrival and service times. Proc. Cambridge Philos. Soc. 58 497--520.
• Mairesse, J. and Prabhakar, B. (1999). On the existence of fixed points for the $\cdot/GI/1$ queue. LIAFA Research Report 99/25, Université Paris 7.
• Martin, J. (2002). Large tandem queueing networks with blocking. Queueing Systems Theory Appl. 41 45--72.
• Mountford, T. and Prabhakar, B. (1995). On the weak convergence of departures from an infinite sequence of $\cdot/M/1$ queues. Ann. Appl. Probab. 5 121--127.
• Prabhakar, B. (2003). The attractiveness of the fixed points of a $\cdot/GI/1$ queue. Ann. Probab. 31 2237--2269.
• Rudin, W. (1991). Functional Analysis, 2nd ed. McGraw-Hill, New York.
• Stoyan, D. (1984). Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.
• Whitt, W. (1992). Uniform conditional stochastic order. J. Appl. Probab. 17 112--123.