Abstract
We consider the following Type of problems. Calls arrive at a queue of capacity K (which is called the primary queue), and attempt to get served by a single server. If upon arrival, the queue is full and the server is busy, the new arriving call moves into an infinite capacity orbit, from which it makes new attempts to reach the primary queue, until it finds it non-full (or it finds the server idle). If the queue is not full upon arrival, then the call (customer) waits in line, and will be served according to the FIFO order. If λ is the arrival rate (average number per time unit) of calls and μ is one over the expected service time in the facility, it is well known that μ > λ is not always sufficient for stability. The aim of this paper is to provide general conditions under which it is a sufficient condition. In particular, (i) we derive conditions for Harris ergodicity and obtain bounds for the rate of convergence to the steady state and large deviations results, in the case that the inter-arrival times, retrial times and service times are independent i.i.d. sequences and the retrial times are exponentially distributed; (ii) we establish conditions for strong coupling convergence to a stationary regime when either service times are general stationary ergodic (no independence assumption), and inter-arrival and retrial times are i.i.d. exponentially distributed; or when inter-arrival times are general stationary ergodic, and service and retrial times are i.i.d. exponentially distributed; (iii) we obtain conditions for the existence of uniform exponential bounds of the queue length process under some rather broad conditions on the retrial process. We finally present conditions for boundedness in distribution for the case of nonpatient (or non persistent) customers.
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References
A.A. Borovkov, Stochastic Processes in Queueing Theory (Springer, 1976).
A.A. Borovkov, Asymptotic Methods in Queueing Theory (Wiley, 1984).
A.A. Borovkov, Ergodicity and stability of multidimensional Markov chains, SIAM Theory Probab. Appl. 35(3) (1989) 542–546.
A.A. Borovkov, Ergodicity and stability of stochastic processes, to appear (in Russian).
A.A. Borovkov, On boundedness of stochastic sequences and estimates of their moments, Ann. Appl. Probab., to appear.
A.A. Borovkov and S.G. Foss, Stochastically recursive sequences and their generalizations, Siberian Adv. Math. 2(1) (1992) 16–81.
J.W. Cohen, Basic problems of telephone traffic theory and the influence of repeated calls, Philips Telecommun. Rev. 18(2) (1957) 49–100.
N. Deul, Stationary conditions for multi-server queueing systems with repeated calls, Elektr. Informationsverarbeit. Kybernet. 16 (1980) 607–613.
G.I. Falin, On sufficient conditions for ergodicity of multichannel queueing systems with repeated calls, Adv. in Appl. Probab. 16 (1984) 447–448.
G.I. Falin, The ergodicity of multilinear queueing systems with repeated calls, Tekhnicheskaya Kibernetika 4 (1986) 63–68.
G.I. Falin, Single-line repeated orders queueing systems, Math. Operat. Forsch. Statist. Ser. Optim. 5 (1986) 649–667.
G.I. Falin, On ergodicity of multichannel queueing systems with repeated calls, Sov. J. Comput. Statistik, Optimization 5 (1986) 649–667.
G.I. Falin, Estimation of error in approximation of countable Markov chains associated with models of repeated calls, Vestnik Moscov. Univ. Ser. I, Mat. Mekh. 2 (1987) 12–15.
G.I. Falin, A survey of retrial queues, Queueing Systems 5 (1990) 127–167.
B. Hajek, Hitting-time and occupation-time bounds implied by drift analysis with applications, Adv. in Appl. Probab. 14 (1982) 502–525.
T. Hanschke, Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts, J. Appl. Probab. 24 (1987) 486–494.
J. Keilson, J. Cozzolino and H. Young, A service system with unfilled requests repeated, Oper. Res. 16 (1968) 1126–1137.
H.-L. Liang and V.G. Kulkarni, Stability condition for a single-server retrial queue, Adv. in Appl. Probab. 25(3) (1993) 690–701.
F.M. Spieksma and R.L. Tweedie, Strengthening ergodicity to geometric ergodicity of Markov chains, Stoch. Models 10(1) (1994) 45–74.
R.L. Tweedie, Criteria for classifying general Markov chains, Adv. in Appl. Probab. 8 (1976) 737–771.
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Altman, E., Borovkov, A.A. On the stability of retrial queues. Queueing Systems 26, 343–363 (1997). https://doi.org/10.1023/A:1019193527040
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DOI: https://doi.org/10.1023/A:1019193527040