Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T01:36:48.111Z Has data issue: false hasContentIssue false
  • English
  • Français

On queues in discrete regenerative environments, with application to the second of two queues in series

Published online by Cambridge University Press:  01 July 2016

K. Balagopal
K. Balagopal*
Affiliation:
Indian Statistical Institute, New Delhi
*
∗Postal address: Statistical Quality Control and Operations Research Unit, Indian Statistical Institute, 7 S.J.S. Sansanwal Marg, New Delhi 110 029, India.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Un be the time between the nth and (n + 1)th arrivals to a single-server queuing system, and Vn the nth arrival's service time. There are quite a few models in which {Un, Vn, n ≥ 1} is a regenerative sequence. In this paper, some light and heavy traffic limit theorems are proved solely under this assumption; some of the light traffic results, and all the heavy traffic results, are new for two such models treated earlier by the author; and all the results are new for the semi-Markov queuing model.

In the last three sections, the results are applied to a single-server queue whose input is the output of a G/G/1 queue functioning in light traffic.

Keywords

DISCRETE-PARAMETER REGENERATIVE PROCESSESIDLE TIMEWAITING TIMEDEPARTURESSERIES QUEUESSTRONG LIMIT AND CENTRAL LIMIT THEOREMSLIGHT TRAFFIC AND HEAVY TRAFFIC LIMIT THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 4 , December 1979 , pp. 851 - 869
Copyright
Copyright © Applied Probability Trust 1979 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Arjas, E. (1972) On a fundamental identity in the theory of semi-Markov processes. Adv. Appl. Prob. 4, 258–270.Google Scholar
2. Arjas, E. (1972) On the use of a fundamental identity in the theory of semi-Markov queues. Adv. Appl. Prob. 4, 271–284.Google Scholar
3. Balagopal, K. (1978) Limit Theory For Some Queuing and Storage Models With Stochastic Environments, Ph.D. thesis submitted to Kakatiya University, Warangal, India.Google Scholar
4. Balagopal, K. (1977) Limit theorems for the single server queue with non-preemptive service interference. Opsearch 14, 244–263.Google Scholar
5. Balagopal, K. (1977) A generalised random walk in the theory of queues and dams. Sankhyā A39, 264–277.Google Scholar
6. Balagopal, K. (1978) A conservation law for utilisation in queues, Opsearch 15, 12–21.Google Scholar
7. Boxma, O. J. (1975) The single-server queue with random service output. J. Appl. Prob. 12, 763–778.Google Scholar
8. Chung, K. L. (1967) Markov Chains With Stationary Transition Probabilities. Springer-Verlag, New York.Google Scholar
9. Chung, K. L. and Fuchs, W. H. J. (1951) On the distribution of values of sums of random variables. Mem. Amer. Math. Soc. 6.CrossRefGoogle Scholar
10. Çinlar, E. (1967) Time-dependence of queues with semi-Markovian service times. J. Appl. Prob. 4, 356–364.CrossRefGoogle Scholar
11. Çinlar, E. (1967) Queues with semi-Markovian arrivals. J. Appl. Prob. 4, 365–379.Google Scholar
12. Erdös, P. and Kac, M. (1946) On certain limit theorems of the theory of probability. Bull. Amer. Math. Soc. 52, 292–302.Google Scholar
13. Feller, W. (1949) Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67, 98–119.CrossRefGoogle Scholar
14. Feller, W. (1966) An Introduction to Probability Theory and its Applications, II. Wiley Eastern, New Delhi.Google Scholar
15. Grinstein, J. and Rubinovitch, M. (1974) Queues with random service output: the case of Poisson arrivals. J. Appl. Prob. 11, 771–784.Google Scholar
16. Hooke, J. A. (1970) On some limit theorems for the GI/G/1 queue. J. Appl. Prob. 7, 634–640.Google Scholar
17. Iglehart, D. L. (1971) Functional limit theorems for the queue GI/G/1 in light traffic. Adv. Appl. Prob. 3, 269–281.Google Scholar
18. Loynes, R. M. (1962) The stability of a queue with non-independent interarrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
19. Neuts, M. F. (1966) The single-server queue with Poisson input and semi-Markovian service times. J. Appl. Prob. 3, 202–230.Google Scholar
20. Neuts, M. F. (1971) A queue subject to extraneous phase changes. Adv. Appl. Prob. 3, 78–119.Google Scholar
21. Purdue, P. (1974) The M/M/1 queue in a Markovian environment. Operat. Res. 22, 562–569.CrossRefGoogle Scholar
22. Purdue, P. (1975) A queue with Poisson input and semi-Markov service times: busy period analysis. J. Appl. Prob. 12, 353–357.Google Scholar
23. Smith, W. L. (1955) Regenerative stochastic processes. Proc. R. Soc. London A232, 6–31.Google Scholar
24. Stidham, S. (1972) Regenerative stochastic processes in the theory of queues, with application to alternating priority queue. Adv. Appl. Prob. 4, 542–577.Google Scholar
25. Takács, L. (1956) On a probability problem arising in the theory of counters. Proc. Camb. Phil. Soc. 32, 488–498.Google Scholar
26. Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar
27. Takács, L. (1976) A Banach space of matrix functions and its application in the theory of queues. Sankhyá A 38, 201–211.Google Scholar
28. Yechiali, U. (1973) A queuing type birth and death process defined on a continuous-time Markov chain. Operat. Res. 21, 604–609.CrossRefGoogle Scholar
29. Yechiali, U. and Naor, P. (1971) Queuing problems with heterogeneous arrivals and services. Operat. Res. 19, 722–734.CrossRefGoogle Scholar