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A state-dependent queueing system with asymptotic logarithmic distribution. (English) Zbl 1388.60151

Summary: A Markovian single-server queueing model with Poisson arrivals and state-dependent service rates, characterized by a logarithmic steady-state distribution, is considered. The Laplace transforms of the transition probabilities and of the densities of the first-passage time to zero are explicitly evaluated. The performance measures are compared with those ones of the well-known \(\mathrm{M}/\mathrm{M}/1\) queueing system. Finally, the effect of catastrophes is introduced in the model and the steady-state distribution, the asymptotic moments and the first-visit time density to zero state are determined.

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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[1] Abate, J.; Whitt, W., Transient behavior of the \(M / M / 1\) queue via Laplace transforms, Adv. in Appl. Probab., 20, 145-178 (1988) · Zbl 0638.60097
[2] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1972), Dover: Dover New York · Zbl 0543.33001
[3] Bailey, N. T.J., A continuous time treatment of a simple queue using generating functions, J. Roy. Statist. Soc. Ser. B (Methodological), 16, 2, 288-291 (1954) · Zbl 0058.34801
[4] Brockwell, P. J.; Gani, J.; Resnick, S. I., Birth, immigration and catastrophe processes, Adv. in Appl. Probab., 14, 709-731 (1982) · Zbl 0496.92007
[5] Chan, J.; Conolly, B., Comparative effectiveness of certain queueing systems with adaptive demand and service mechanisms, Comput. Oper. Res., 5, 3, 187-196 (1978)
[6] Chang, I.; Krinik, A.; Swift, R. J., Birth-multiple catastrophe processes, J. Statist. Plann. Inference, 137, 1544-1559 (2007) · Zbl 1114.60064
[7] Chao, X.; Zheng, Y., Transient and equilibrium analysis of an immigration birth-death process with total catastrophes, Probab. Engrg. Inform. Sci., 17, 1, 83-106 (2003) · Zbl 1019.60079
[8] Chen, A.; Renshaw, E., The \(M / M / 1\) queue with mass exodus and mass arrivals when empty, J. Appl. Probab., 34, 197-207 (1997) · Zbl 0876.60079
[9] Cohen, J. W., The Single Server Queue (1982), North-Holland Publishing Company: North-Holland Publishing Company Amsterdam-London · Zbl 0481.60003
[10] Conolly, B. W., The generalised state-dependent Erlangian queue: the busy period, J. Appl. Probab., 11, 618-623 (1974) · Zbl 0288.60084
[11] Conolly, B. W., Generalized state-dependent Erlangian queues: speculations about calculating measures of effectiveness, J. Appl. Probab., 12, 358-363 (1975) · Zbl 0304.60051
[12] Conolly, B. W., More transient results for generalised state-dependent Erlangian queues, Adv. in Appl. Probab., 15, 688-690 (1983) · Zbl 0515.60095
[13] Conolly, B. W.; Langaris, C., On a new formula for the transient state probabilities for \(M / M / 1\) queues and computational implications, J. Appl. Probab., 30, 237-246 (1993) · Zbl 0768.60086
[14] Daly, F.; Lefevre, C.; Utev, S., Stein’s method and stochastic orderings, Adv. in Appl. Probab., 44, 343-372 (2012) · Zbl 1276.62015
[15] Di Crescenzo, A.; Giorno, V.; Nobile, A. G.; Ricciardi, L. M., On the \(M / M / 1\) queue with catastrophes and its continuous approximation, Queueing Syst., 43, 329-347 (2003) · Zbl 1016.60080
[16] Di Crescenzo, A.; Giorno, V.; Nobile, A. G.; Ricciardi, L. M., A note on birth-death processes with catastrophes, Statist. Probab. Lett., 78, 2248-2257 (2008) · Zbl 1283.60110
[17] Di Crescenzo, A.; Giorno, V.; Krishna Kumar, B.; Nobile, A. G., A double-ended queue with catastrophes and repairs, and a jump-diffusion approximation, Methodol. Comput. Appl. Probab., 14, 937-954 (2012) · Zbl 1270.60080
[18] Economou, A.; Fakinos, D., A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes, European J. Oper. Res., 149, 625-640 (2003) · Zbl 1030.60069
[19] Economou, A.; Fakinos, D., Alternative approaches for the transient analysis of Markov chains with catastrophes, J. Stat. Theory Pract., 2, 183-197 (2008) · Zbl 1420.60099
[20] Giorno, V.; Negri, C.; Nobile, A. G., A solvable model for a finite capacity queueing system, J. Appl. Probab., 22, 903-911 (1985) · Zbl 0576.60089
[21] Gradshteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series, and Products (2007), Academic Press: Academic Press Amsterdam · Zbl 1208.65001
[22] Gupta, U. C.; Srinivasa Rao, T. S.S., On the analysis of single server finite queue with state dependent arrival and service processes: \(M(n) / G(n) / 1 / k\), OR Spectrum, 20, 83-89 (1998) · Zbl 0904.90059
[23] Hadidi, N., Busy periods of queues with state dependent arrival and service rates, J. Appl. Probab., 11, 842-848 (1974) · Zbl 0309.60063
[24] Hadidi, N., A queueing model with variable arrival rates, Period. Math. Hungar., 6, 1, 39-47 (1975) · Zbl 0264.60074
[25] Hadidi, N., Queues with partial correlation, SIAM J. Appl. Math., 40, 3, 467-475 (1981) · Zbl 0462.60093
[26] Jouini, O.; Dallery, Y., Moments of first passage times in general birth-death processes, Math. Methods Oper. Res., 68, 49-76 (2008) · Zbl 1158.60372
[27] Karlin, S.; McGregor, J., The classification of birth and death processes, Trans. Amer. Math. Soc., 86, 366-400 (1957) · Zbl 0091.13802
[28] Kijima, M., Markov Processes for Stochastic Modeling (1997), Chapman & Hall: Chapman & Hall London · Zbl 0866.60056
[29] Krishna Kumar, B.; Arivudainambi, D., Transient solution of an \(M / M / 1\) queue with catastrophes, Comput. Math. Appl., 40, 1233-1240 (2000) · Zbl 0962.60096
[30] Krishna Kumar, B.; Krishnamoorthy, A.; Pavai Madheswari, S.; Sadiq Basha, S., Transient analysis of a single server queue with catastrophes, failures and repairs, Queueing Syst., 56, 133-141 (2007) · Zbl 1124.60073
[31] Kyriakidis, E. G., The transient probabilities of the simple immigration-catastrophe process, Math. Sci., 26, 56-58 (2001) · Zbl 1113.92318
[32] Kyriakidis, E. G., Transient solution for a simple immigration birth-death process, Probab. Engrg. Inform. Sci., 18, 233-236 (2004) · Zbl 1049.60074
[33] Natvig, B., On the transient state probabilities for a queueing model where potential customers are discouraged by queue length, J. Appl. Probab., 11, 345-354 (1974) · Zbl 0278.60076
[34] Natvig, B., On a queuing model where potential customers are discouraged by queue length, Scand. J. Stat., 2, 34-42 (1975) · Zbl 0296.60069
[35] Pakes, A. G., Killing and resurrection of Markov processes, Commun. Stat., Stoch. Models, 13, 255-269 (1997) · Zbl 0880.60066
[36] Parthasarathy, P. R.; Selvaraju, N., Transient analysis of a queue where potential customers are discouraged by queue length, Math. Probl. Eng., 7, 5, 433-454 (2001) · Zbl 1008.60098
[37] Renshaw, E.; Chen, A., birth-death processes with mass annihilation and state-dependent immigration, Stoch. Models, 13, 239-253 (1997) · Zbl 0880.60088
[38] Ross, N., Fundamentals of Stein’s method, Probab. Surv., 11, 210-293 (2011) · Zbl 1245.60033
[39] Sharma, O. P.; Bunday, B. D., A simple formula for the transient state probabilities of an \(M / M / 1 / \infty\) queue, Optimization, 40, 79-84 (1997) · Zbl 0876.60076
[40] Sudhesh, R., Transient analysis of a queue with system disasters and customer impatience, Queueing Syst., 66, 95-105 (2010) · Zbl 1197.60087
[41] Sundt, B.; Vernic, R., Recursions for Convolutions and Compound Distributions with Insurance Applications, EAA Lecture Notes (2009), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 1175.60004
[42] Swift, R. J., Transient probabilities for a simple birth-death-immigration process under the influence of total catastrophes, Int. J. Math. Sci., 25, 689-692 (2001) · Zbl 0989.60080
[43] Upadhye, N. S.; Čekanavičius, V.; Vellaisamy, P., On Stein operators for discrete approximations, Bernoulli, 23, 4A, 2828-2859 (2017) · Zbl 1390.60094
[44] Van Doorn, E. A., The transient state probabilities for a queueing model where potential customers are discouraged by queue length, J. Appl. Probab., 18, 499-506 (1981) · Zbl 0457.60070
[45] Whitt, W., Comparing counting processes and queues, Adv. in Appl. Probab., 13, 207-220 (1981) · Zbl 0449.60064
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