Transient analysis of queues with heterogeneous arrivals. (English) Zbl 0806.60083
Summary: We consider a discrete time queueing model where the time axis is divided into time slots of unit length. The model satisfies the following assumptions.
(i) An event is either an arrival of type \(i\) of batch size \(b_ i\), \(i = 1, \dots, r\), with probability \(\alpha_ i\) or is a depature of a single customer with probability \(\gamma\) or zero depending on whether the queue is busy or empty.
(ii) No more than one event can occur in a slot, therefore the probability that neither an arrival nor a departure occurs in a slot is \(1 - \gamma - \sum_ i \alpha_ i\) or \(1-\sum_ i \alpha_ i\) according as the queue is busy or empty.
(iii) Events in different slots are independent.
Using a lattice path representation in higher dimensional space we derive the time dependent joint distribution of the number of arrivals of various types and the number of completed services. The distribution for the corresponding continuous time model is found by using weak convergence.
(i) An event is either an arrival of type \(i\) of batch size \(b_ i\), \(i = 1, \dots, r\), with probability \(\alpha_ i\) or is a depature of a single customer with probability \(\gamma\) or zero depending on whether the queue is busy or empty.
(ii) No more than one event can occur in a slot, therefore the probability that neither an arrival nor a departure occurs in a slot is \(1 - \gamma - \sum_ i \alpha_ i\) or \(1-\sum_ i \alpha_ i\) according as the queue is busy or empty.
(iii) Events in different slots are independent.
Using a lattice path representation in higher dimensional space we derive the time dependent joint distribution of the number of arrivals of various types and the number of completed services. The distribution for the corresponding continuous time model is found by using weak convergence.
MSC:
| 60K25 | Queueing theory (aspects of probability theory) |
| 05A15 | Exact enumeration problems, generating functions |
| 90B22 | Queues and service in operations research |
Keywords:
lattice path enumeration; discrete time queueing model; lattice path representation; weak convergenceReferences:
| [1] | W. B?hm,Markovian Queueing Systems in Discrete Time, Mathematical Systems in Economics, Vol. 137 (Hain, Frankfurt/Main, 1993). |
| [2] | W. B?hm, and S.G. Mohanty, On discrete time MarkovianN-policy queues involving batches, Sanky?, Ser. A 56 (1994) 1-20. |
| [3] | W. B?hm and S.G. Mohanty, Transient analysis ofM/M/1 queues in discrete time with general server vacations, J. Appl. Prob. (1992), to appear. |
| [4] | W. B?hm and S.G. Mohanty, The transient solution ofM/M/1 queues under (M,N)-policy. A combinatorial approach, J. Statist. Planning Inference 34 (1993) 23-33. · Zbl 0777.05004 · doi:10.1016/0378-3758(93)90031-Z |
| [5] | W. Feller,An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. (Wiley, New York, 1968). · Zbl 0155.23101 |
| [6] | H.W. Gould, Some generalizations of Vandermonde’s convolution, Amer. Math. Monthly 63 (1956) 84-91. · Zbl 0072.00702 · doi:10.2307/2306429 |
| [7] | Kanwar Sen, J.L. Jain and J.M. Gupta, Lattice path approach to transient solution ofM/M/1 with (0,K) control policy, J. Statist. Planning Inference 34 (1993) 68-82. · Zbl 0765.60099 |
| [8] | Kanwar Sen and J.L. Jain, Combinatorial approach to Markovian queues, J. Statist. Planning Inference 34 (1993) 55-67. · Zbl 0766.60117 |
| [9] | Kanwar Sen and J.L. Jain, private communication (1993). |
| [10] | I. Mitrani,Modelling of Computer and Communication Systems (Cambridge University Press, Cambridge, New York, 1987). · Zbl 0648.68015 |
| [11] | S.G. Mohanty, On queues involving batches, J. Appl. Prob. 9 (1972) 430-435. · Zbl 0241.60082 · doi:10.2307/3212812 |
| [12] | S.G. Mohanty,Lattice Path Counting and Applications (Academic Press, New York, 1979). · Zbl 0455.60013 |
| [13] | L. Tak?cs,Combinatorial Methods in the Theory of Stochastic Processes (Wiley, New York, 1967). |
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