Tail asymptotics for a random sign Lindley recursion. (English) Zbl 1186.60023
Summary: We investigate the tail behaviour of the steady-state distribution of a stochastic recursion that generalises Lindley’s recursion. This recursion arises in queueing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley’s recursion and for alternating service systems.
Keywords:
stochastic recursion; Cramér conditions; subexponential distribution; intermediate tail; sharp asymptoticsReferences:
| [1] | Asmussen, S. (2003). Applied Probability and Queues . Springer, New York. · Zbl 1029.60001 |
| [2] | Bertoin, J. and Doney, R. A. (1996). Some asymptotic results for transient random walks. Adv. Appl. Prob. 28 , 207–226. JSTOR: · Zbl 0854.60069 · doi:10.2307/1427918 |
| [3] | Boxma, O. J. and Vlasiou, M. (2007). On queues with service and interarrival times depending on waiting times. Queueing Systems 56 , 121–132. · Zbl 1125.60095 · doi:10.1007/s11134-007-9011-3 |
| [4] | Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72 , 529–557. · Zbl 0577.60019 · doi:10.1007/BF00344720 |
| [5] | Cohen, J. W. (1982). The Single Server Queue . North-Holland, Amsterdam. · Zbl 0481.60003 |
| [6] | Denisov, D. and Zwart, B. (2007). On a theorem of Breiman and a class of random difference equations. J. Appl. Prob. 44 , 1031–1046. · Zbl 1141.60041 · doi:10.1239/jap/1197908822 |
| [7] | Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. 33 ). Springer, Berlin. · Zbl 0873.62116 |
| [8] | Foss, S. and Zachary, S. (2003). The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann. Appl. Prob. 13 , 37–53. · Zbl 1045.60039 · doi:10.1214/aoap/1042765662 |
| [9] | Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1 , 126–166. · Zbl 0724.60076 · doi:10.1214/aoap/1177005985 |
| [10] | Korshunov, D. (1997). On distribution tail of the maximum of a random walk. Stoch. Process. Appl. 72 , 97–103. · Zbl 0942.60018 · doi:10.1016/S0304-4149(97)00060-4 |
| [11] | Lindley, D. V. (1952). The theory of queues with a single server. Proc. Camb. Phil. Soc. 48 , 277–289. · Zbl 0046.35501 · doi:10.1017/S0305004100027638 |
| [12] | Park, B. C., Park, J. Y. and Foley, R. D. (2003). Carousel system performance. J. Appl. Prob. 40 , 602–612. · Zbl 1140.90359 · doi:10.1239/jap/1059060890 |
| [13] | Pitman, E. J. G. (1980). Subexponential distribution functions. J. Austral. Math. Soc. A 29 , 337–347. · Zbl 0425.60012 · doi:10.1017/S1446788700021340 |
| [14] | Veraverbeke, N. (1977). Asymptotic behaviour of Wiener–Hopf factors of a random walk. Stoch. Process. Appl. 5 , 27–37. · Zbl 0353.60073 · doi:10.1016/0304-4149(77)90047-3 |
| [15] | Vlasiou, M. (2007). A non-increasing Lindley-type equation. Queueing Systems 56 , 41–52. · Zbl 1165.90411 · doi:10.1007/s11134-007-9029-6 |
| [16] | Vlasiou, M. and Adan, I. J. B. F. (2005). An alternating service problem. Prob. Eng. Inf. Sci. 19 , 409–426. · Zbl 1336.60180 · doi:10.1017/S0269964805050254 |
| [17] | Vlasiou, M. and Adan, I. J. B. F. (2007). Exact solution to a Lindley-type equation on a bounded support. Operat. Res. Let. 35 , 105–113. · Zbl 1120.60093 · doi:10.1016/j.orl.2005.12.004 |
| [18] | Vlasiou, M., Adan, I. J. B. F. and Wessels, J. (2004). A Lindley-type equation arising from a carousel problem. J. Appl. Prob. 41 , 1171–1181. · Zbl 1062.60095 · doi:10.1239/jap/1101840561 |
| [19] | Whitt, W. (1990). Queues with service times and interarrival times depending linearly and randomly upon waiting times. Queueing Systems 6 , 335–351. · Zbl 0707.60090 · doi:10.1007/BF02411482 |
| [20] | Zachary, S. (2004). A note on Veraverbeke’s theorem. Queueing Systems 46 , 9–14. · Zbl 1056.90040 · doi:10.1023/B:QUES.0000021155.44510.9f |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
