# zbMATH — the first resource for mathematics

The idle period of the finite $$G/M/1$$ queue with an interpretation in risk theory. (English) Zbl 1201.60087
The paper deals with the $$G/M/1$$ queue where the workload (virtual waiting time) $$V_t$$ is bounded by $$1$$: If $$V_t$$ plus the service time of an arriving customer exceeds $$1$$, only $$1-V_t$$ of the service requirement is accepted. The paper focus on the idle period $$I$$, i.e. the duration between the time when the queue becomes empty and the next customer arrives, of this finite queue, because it can be interpreted as the deficit at ruin for a risk reserve process $$R_t$$ in the compound Poisson risk model with a constant behavior strategy: when the risk reserve process reaches level $$1$$ dividends are paid out with constant rate $$1$$, such that $$R_t$$ is constant until the next claim occurs.
In the paper expressions for the LST and the distribution of $$I$$ are given. Two methods are used, which base on sample path analysis: the first method bases on collecting subsequent overshoots over level $$1$$ in the original $$G/M/1$$ workload process and the second method on the observation that the idle period can be seen as the overshoot of the workload process in a dual $$M/G/1$$ queue. The second method is used to construct a modified $$M/G/1$$ process and to define a formula for the distribution of the idle period in the finite $$G/M/1$$ queue with set-up time $$a\in[0,1]$$, where after each busy period an arriving customer has to wait $$a$$ time units until the server is ready to serve it.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 91B30 Risk theory, insurance (MSC2010) 90B22 Queues and service in operations research 68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
Full Text:
##### References:
 [1] Adan, I., Boxma, O., Perry, D.: The G/M/1 queue revisited. Math. Methods Oper. Res. 62(3), 437–452 (2005) · Zbl 1085.60064 [2] Albrecher, H., Kainhofer, R.: Risk theory with a nonlinear dividend barrier. Computing 68(4), 289–311 (2002) · Zbl 1076.91521 [3] Albrecher, H., Hartinger, J., Tichy, R.F.: On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier. Scand. Actuar. J. 2005(2), 103–126 (2005) · Zbl 1092.91036 [4] Asmussen, S.: Applied Probability and Queues. Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. Wiley, Chichester (1987) · Zbl 0624.60098 [5] Asmussen, S.: Ruin Probabilities. World Scientific, Singapore (2000) · Zbl 0960.60003 [6] Avram, F., Usábel, M.: Ruin probabilities and deficit for the renewal risk model with phase-type interarrival times. Astin Bull. 34(2), 315–332 (2004) · Zbl 1274.91244 [7] Bekker, R.: Finite-buffer queues with workload-dependent service and arrival rates. Queueing Syst. 50(2–3), 231–253 (2005) · Zbl 1085.60066 [8] Belzunce, F., Ortega, E.M., Ruiz, J.M.: A note on stochastic comparisons of excess lifetimes of renewal processes. J. Appl. Probab. 38(3), 747–753 (2001) · Zbl 0993.60090 [9] Brill, P.H.: Single-server queues with delay-dependent arrival streams. Probab. Eng. Inf. Sci. 2(2), 231–247 (1988) · Zbl 1134.60388 [10] Brill, P.H.: Level Crossing Methods in Stochastic Models. International Series in Operations Research and Management Science, vol. 123. Springer, Berlin (2008) · Zbl 1157.60003 [11] Brill, P.H., Posner, M.J.M.: Level crossings in point processes applied to queues: Single-server case. Oper. Res. 25, 662–674 (1977) · Zbl 0373.60114 [12] Cohen, J.W.: Extreme value distribution for the M/G/1 and the G/M/1 queueing systems. Ann. Inst. H. Poincare, Sect. B 4, 83–98 (1968) · Zbl 0162.49302 [13] Cohen, J.W.: Single-server queue with uniformly bounded virtual waiting time. J. Appl. Probab. 5, 93–122 (1968) · Zbl 0164.48103 [14] Cohen, J.W.: On up- an down-crossings. J. Appl. Probab. 14, 405–410 (1977) · Zbl 0365.60108 [15] Cohen, J.W.: The Single Server Queue. North-Holland Series in Applied Mathematics and Mechanics, vol. 8. North-Holland, Amsterdam (1982) · Zbl 0481.60003 [16] Daley, D.J.: Single-server queueing systems with uniformly limited queueing time. J. Aust. Math. Soc. 4, 489–505 (1964) · Zbl 0131.16802 [17] Gerber, H.U.: On the probability of ruin in the presence of a linear dividend barrier. Scand. Actuar. J. 1981, 105–115 (1981) · Zbl 0455.62086 [18] Gerber, H.U., Goovaerts, M.J., Kaas, R.: On the probability and severity of ruin. ASTIN Bull. Int. Actuar. Assoc.–Bruss., Belg. 17(2), 151–164 (1987) [19] Hu, J.Q., Zazanis, M.A.: A sample path analysis of M/GI/1 queues with workload restrictions. Queueing Syst. 14(1–2), 203–213 (1993) · Zbl 0780.60091 [20] Kaspi, H., Kella, O., Perry, D.: Dam processes with state dependent batch sizes and intermittent production processes with state dependent rates. Queueing Syst. 24(1–4), 37–57 (1996) · Zbl 0874.90075 [21] Kleinrock, L.: Queueing Systems. Vol. I: Theory. Wiley, New York (1975) · Zbl 0334.60045 [22] Li, S., Garrido, J.: On a class of renewal risk models with a constant dividend barrier. Insur. Math. Econ. 35(3), 691–701 (2004) · Zbl 1122.91345 [23] Lin, X.S., Pavlova, K.P.: The compound Poisson risk model with a threshold dividend strategy. Insur. Math. Econ. 38(1), 57–80 (2006) · Zbl 1157.91383 [24] Lin, X.S., Willmot, G.E., Drekic, S.: The classical risk model with a constant dividend barrier: Analysis of the Gerber–Shiu discounted penalty function. Insur. Math. Econ. 33(3), 551–566 (2003) · Zbl 1103.91369 [25] Minh, D.L.: The GI/G/1 queue with uniformly limited virtual waiting times; the finite dam. Adv. Appl. Probab. 12, 501–516 (1980) · Zbl 0425.60085 [26] Perry, D., Stadje, W.: A controlled M/G/1 workload process with an application to perishable inventory systems. Math. Methods Oper. Res. 64(3), 415–428 (2006) · Zbl 1131.90017 [27] Perry, D., Stadje, W., Zacks, S.: Busy period analysis for M/G/1 and G/M/1 type queues with restricted accessibility. Oper. Res. Lett. 27(4), 163–174 (2000) · Zbl 1096.90517 [28] Perry, D., Stadje, W., Zacks, S.: The M/G/1 queue with finite workload capacity. Queueing Syst. 39(1), 7–22 (2001) · Zbl 1002.90014 [29] Perry, D., Stadje, W., Zacks, S.: Hitting and ruin probabilities for compound Poisson processes and the cycle maximum of the M/G/1 queue. Stoch. Models 18(4), 553–564 (2002) · Zbl 1013.60070 [30] Prabhu, N.U.: Stochastic Storage Processes. Queues, Insurance Risk, and Dams. Applications of Mathematics, vol. 15. Springer, Berlin (1980) · Zbl 0453.60094 [31] Roes, P.B.M.: The finite dam. J. Appl. Probab. 7, 316–326 (1970) · Zbl 0214.18802 [32] Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic Processes for Insurance and Finance. Wiley, Chichester (1999) · Zbl 0940.60005 [33] Ross, S.M., Seshadri, S.: Hitting time in an M/G/1 queue. J. Appl. Probab. 36(3), 934–940 (1999) · Zbl 0953.60082 [34] Takács, L.: Introduction to the Theory of Queues. University Texts in the Mathematical Sciences. Oxford University Press, New York (1962) · Zbl 0106.33502 [35] Takács, L.: Application of Ballot theorems in the theory of queues. In: Proc. Symp. Congestion Theory, pp. 337–393. Chapel Hill (1965) · Zbl 0203.18402 [36] Takács, L.: A single-server queue with limited virtual waiting time. J. Appl. Probab. 11, 612–617 (1974) · Zbl 0303.60098
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.