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Explicit results for the distribution of the number of customers served during a busy period for $M^X/PH/1$ queue
1. | School of Computer Applications, Kalinga Institute of Industrial Technology, Bhubaneswar-751024, India |
2. | Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. Box 17000, Kingston, Ontario, Canada K7K 7B4 |
We give analytically explicit solutions for the distribution of the number of customers served during a busy period for the $ M^X/PH/1 $ queues when initiated with $ m $ customers. When customers arrive in batches, we present the functional equation for the Laplace transform of the number of customers served during a busy period. Applying the Lagrange inversion theorem, we provide a refined result to this functional equation. From a phase-type service distribution, we obtain the distribution of the number of customers served during a busy period for various special cases such as exponential, Erlang-k, generalized Erlang, hyperexponential, Coxian, and interrupted Poisson process. The results are exact, rapid and vigorous, owing to the clarity of the expressions. Moreover, we also consider computational results for several service-time distributions using our method. Phase-type distributions can approximate any non-negative valued distribution arbitrarily close, making them a useful practical stochastic modelling tool. These distributions have eloquent properties which make them beneficial in the computation of performance models.
References:
[1] |
J. Abate and W. Whitt,
The fourier-series method for inverting transforms of probability distributions, Queueing Systems, 10 (1992), 5-87.
doi: 10.1007/BF01158520. |
[2] |
J. Abate and W. Whitt,
Numerical inversion of probability generating functions, Operations Research Letters, 12 (1992), 245-251.
doi: 10.1016/0167-6377(92)90050-D. |
[3] |
J. Abate and W. Whitt,
Solving probability transform functional equations for numerical inversion, Operations Research Letters, 12 (1992), 275-281.
doi: 10.1016/0167-6377(92)90085-H. |
[4] |
S. Asmussen and H. Albrecher, Ruin Probabilities, vol. 14, Advanced Series on Statistical Science & Applied Probability, Hackensack, NJ, 2010.
doi: 10.1142/9789814282536. |
[5] |
E. Borel,
Sur l'emploi du théoreme de Bernoulli pour faciliter le calcul d'une infinité de coefficients. Application au probleme de l'attentea un guichet, CR Acad. Sci. Paris, 214 (1942), 452-456.
|
[6] |
L. Breuer and D. Baum, An Introduction to Queueing Theory: And Matrix-Analytic Methods, Springer Science & Business Media, 2005.
doi: 10.1007/1-4020-3631-0. |
[7] |
M. L. Chaudhry and V. Goswami, Analytically explicit results for the distribution of the number of customers served during a busy period for special cases of the $M/G/1$ queue, Journal of Probability and Statistics, 2019 (2019), Art. ID 7398658, 15 pp.
doi: 10.1155/2019/7398658. |
[8] | |
[9] |
P. C. Consul and F. Famoye, Lagrangian Probability Distributions, Springer, 2006. |
[10] |
P. C. Consul and L. R. Shenton,
Use of lagrange expansion for generating discrete generalized probability distributions, SIAM Journal on Applied Mathematics, 23 (1972), 239-248.
doi: 10.1137/0123026. |
[11] |
D. R. Cox,
Some statistical methods connected with series of events, Journal of the Royal Statistical Society: Series B (Methodological), 17 (1955), 129-157.
doi: 10.1111/j.2517-6161.1955.tb00188.x. |
[12] |
A. K. Erlang,
Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges, Post Office Electrical Engineer's Journal, 10 (1917), 189-197.
|
[13] |
G. Falin,
Functioning under nonsteady conditions of a single-channel system with group arrival of requests and repeated calls, Ukrainian Mathematical Journal, 33 (1981), 429-432.
doi: 10.1007/BF01085753. |
[14] |
W. Fischer and K. Meier-Hellstern,
The Markov-modulated Poisson process (MMPP) cookbook, Performance Evaluation, 18 (1993), 149-171.
doi: 10.1016/0166-5316(93)90035-S. |
[15] |
F. A. Haight,
A distribution analogous to the borel-tanner, Biometrika, 48 (1961), 167-173.
doi: 10.1093/biomet/48.1-2.167. |
[16] |
D. P. Heyman,
An approximation for the busy period of the ${M/G/1}$ queue using a diffusion model, Journal of Applied Probability, 11 (1974), 159-169.
doi: 10.2307/3212592. |
[17] |
D. G. Kendall,
Some problems in the theory of dams, Journal of the Royal Statistical Society. Series B (Methodological), 19 (1957), 207-233.
doi: 10.1111/j.2517-6161.1957.tb00257.x. |
[18] |
J. Kim,
Busy period distribution of a batch arrival retrial queue, Communications of the Korean Mathematical Society, 32 (2017), 425-433.
doi: 10.4134/CKMS.c160106. |
[19] |
L. Kleinrock, Queueing Systems, vol. 1, Wiley, New York, 1975. |
[20] |
G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, 1999.
doi: 10.1137/1.9780898719734. |
[21] |
J. Medhi, Stochastic Models in Queueing Theory, Academic Press, Amsterdam, 2003.
![]() ![]() |
[22] |
M. F. Neuts,
Computational uses of the method of phases in the theory of queues, Computers & Mathematics with Applications, 1 (1975), 151-166.
doi: 10.1016/0898-1221(75)90015-2. |
[23] |
M. F. Neuts,
Matrix-geometric solutions in stochastic models: An algorithmic approach, Bull. Amer. Math. Soc, 8 (1983), 97-99.
doi: 10.1090/S0273-0979-1983-15095-4. |
[24] |
N. U. Prabhu,
Some results for the queue with Poisson arrivals, Journal of the Royal Statistical Society: Series B (Methodological), 22 (1960), 104-107.
doi: 10.1111/j.2517-6161.1960.tb00357.x. |
[25] |
N. U. Prabhu, Queues and Inventories, John Wiley & Sons, 1965. |
[26] |
J. F. Shortle, J. M. Thompson, D. Gross and C. M. Harris, Fundamentals of Queueing Theory, Fifth Edition, John Wiley & Sons, 2018.
doi: 10.1002/9781119453765. |
[27] |
W. J. Stewart, Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling, Princeton University Press, 2009. |
[28] |
L. Takács, Combinatorial Methods in the Theory of Stochastic Processes, Wiley, New York, 1967.
![]() ![]() |
[29] |
J. C. Tanner,
A problem of interference between two queues, Biometrika, 40 (1953), 58-69.
doi: 10.1093/biomet/40.1-2.58. |
show all references
References:
[1] |
J. Abate and W. Whitt,
The fourier-series method for inverting transforms of probability distributions, Queueing Systems, 10 (1992), 5-87.
doi: 10.1007/BF01158520. |
[2] |
J. Abate and W. Whitt,
Numerical inversion of probability generating functions, Operations Research Letters, 12 (1992), 245-251.
doi: 10.1016/0167-6377(92)90050-D. |
[3] |
J. Abate and W. Whitt,
Solving probability transform functional equations for numerical inversion, Operations Research Letters, 12 (1992), 275-281.
doi: 10.1016/0167-6377(92)90085-H. |
[4] |
S. Asmussen and H. Albrecher, Ruin Probabilities, vol. 14, Advanced Series on Statistical Science & Applied Probability, Hackensack, NJ, 2010.
doi: 10.1142/9789814282536. |
[5] |
E. Borel,
Sur l'emploi du théoreme de Bernoulli pour faciliter le calcul d'une infinité de coefficients. Application au probleme de l'attentea un guichet, CR Acad. Sci. Paris, 214 (1942), 452-456.
|
[6] |
L. Breuer and D. Baum, An Introduction to Queueing Theory: And Matrix-Analytic Methods, Springer Science & Business Media, 2005.
doi: 10.1007/1-4020-3631-0. |
[7] |
M. L. Chaudhry and V. Goswami, Analytically explicit results for the distribution of the number of customers served during a busy period for special cases of the $M/G/1$ queue, Journal of Probability and Statistics, 2019 (2019), Art. ID 7398658, 15 pp.
doi: 10.1155/2019/7398658. |
[8] | |
[9] |
P. C. Consul and F. Famoye, Lagrangian Probability Distributions, Springer, 2006. |
[10] |
P. C. Consul and L. R. Shenton,
Use of lagrange expansion for generating discrete generalized probability distributions, SIAM Journal on Applied Mathematics, 23 (1972), 239-248.
doi: 10.1137/0123026. |
[11] |
D. R. Cox,
Some statistical methods connected with series of events, Journal of the Royal Statistical Society: Series B (Methodological), 17 (1955), 129-157.
doi: 10.1111/j.2517-6161.1955.tb00188.x. |
[12] |
A. K. Erlang,
Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges, Post Office Electrical Engineer's Journal, 10 (1917), 189-197.
|
[13] |
G. Falin,
Functioning under nonsteady conditions of a single-channel system with group arrival of requests and repeated calls, Ukrainian Mathematical Journal, 33 (1981), 429-432.
doi: 10.1007/BF01085753. |
[14] |
W. Fischer and K. Meier-Hellstern,
The Markov-modulated Poisson process (MMPP) cookbook, Performance Evaluation, 18 (1993), 149-171.
doi: 10.1016/0166-5316(93)90035-S. |
[15] |
F. A. Haight,
A distribution analogous to the borel-tanner, Biometrika, 48 (1961), 167-173.
doi: 10.1093/biomet/48.1-2.167. |
[16] |
D. P. Heyman,
An approximation for the busy period of the ${M/G/1}$ queue using a diffusion model, Journal of Applied Probability, 11 (1974), 159-169.
doi: 10.2307/3212592. |
[17] |
D. G. Kendall,
Some problems in the theory of dams, Journal of the Royal Statistical Society. Series B (Methodological), 19 (1957), 207-233.
doi: 10.1111/j.2517-6161.1957.tb00257.x. |
[18] |
J. Kim,
Busy period distribution of a batch arrival retrial queue, Communications of the Korean Mathematical Society, 32 (2017), 425-433.
doi: 10.4134/CKMS.c160106. |
[19] |
L. Kleinrock, Queueing Systems, vol. 1, Wiley, New York, 1975. |
[20] |
G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, 1999.
doi: 10.1137/1.9780898719734. |
[21] |
J. Medhi, Stochastic Models in Queueing Theory, Academic Press, Amsterdam, 2003.
![]() ![]() |
[22] |
M. F. Neuts,
Computational uses of the method of phases in the theory of queues, Computers & Mathematics with Applications, 1 (1975), 151-166.
doi: 10.1016/0898-1221(75)90015-2. |
[23] |
M. F. Neuts,
Matrix-geometric solutions in stochastic models: An algorithmic approach, Bull. Amer. Math. Soc, 8 (1983), 97-99.
doi: 10.1090/S0273-0979-1983-15095-4. |
[24] |
N. U. Prabhu,
Some results for the queue with Poisson arrivals, Journal of the Royal Statistical Society: Series B (Methodological), 22 (1960), 104-107.
doi: 10.1111/j.2517-6161.1960.tb00357.x. |
[25] |
N. U. Prabhu, Queues and Inventories, John Wiley & Sons, 1965. |
[26] |
J. F. Shortle, J. M. Thompson, D. Gross and C. M. Harris, Fundamentals of Queueing Theory, Fifth Edition, John Wiley & Sons, 2018.
doi: 10.1002/9781119453765. |
[27] |
W. J. Stewart, Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling, Princeton University Press, 2009. |
[28] |
L. Takács, Combinatorial Methods in the Theory of Stochastic Processes, Wiley, New York, 1967.
![]() ![]() |
[29] |
J. C. Tanner,
A problem of interference between two queues, Biometrika, 40 (1953), 58-69.
doi: 10.1093/biomet/40.1-2.58. |







Batch size distribution | ||
Geometric | ||
Arbitrary |
Batch size distribution | ||
Geometric | ||
Arbitrary |
Batch size distribution | ||
Geometric | ||
Arbitrary |
Batch size distribution | ||
Geometric | ||
Arbitrary |
Batch size distribution | ||
Geometric | ||
Arbitrary |
Batch size distribution | ||
Geometric | ||
Arbitrary |
Batch size distribution | ||
Geometric | ||
Arbitrary |
Batch size distribution | ||
Geometric | ||
Arbitrary |
Batch size distribution | ||
Geometric | ||
Arbitrary |
Batch size distribution | ||
Geometric | ||
Arbitrary |
Batch size distribution | ||
Geometric | ||
Arbitrary |
Batch size distribution | ||
Geometric | ||
Arbitrary |
Geometric batch size |
Arbitrary batch size |
|||||
1 | 0.869565 | 0.769231 | 0.689655 | 0.864865 | 0.761905 | 0.680851 |
2 | 0.059176 | 0.08193 | 0.088565 | 0.040432 | 0.055286 | 0.059178 |
3 | 0.028637 | 0.042662 | 0.047178 | 0.056233 | 0.071207 | 0.070724 |
4 | 0.015533 | 0.025829 | 0.029730 | 0.012702 | 0.024379 | 0.028500 |
5 | 0.009138 | 0.017213 | 0.020736 | 0.012009 | 0.023001 | 0.026226 |
6 | 0.005694 | 0.012222 | 0.015438 | 0.004622 | 0.012738 | 0.016531 |
7 | 0.003698 | 0.009070 | 0.012021 | 0.003650 | 0.010773 | 0.014187 |
8 | 0.002478 | 0.006951 | 0.009672 | 0.001825 | 0.007351 | 0.010714 |
9 | 0.001700 | 0.005460 | 0.007977 | 0.001325 | 0.006028 | 0.009135 |
10 | 0.001189 | 0.004373 | 0.006709 | 0.000762 | 0.004546 | 0.007492 |
50 | 0.000000 | 0.000033 | 0.000367 | 0.000000 | 0.000016 | 0.000367 |
100 | 0.000000 | 0.000001 | 0.000072 | 0.000000 | 0.000000 | 0.000062 |
150 | 0.000000 | 0.000000 | 0.000022 | 0.000000 | 0.000000 | 0.000016 |
200 | 0.000000 | 0.000000 | 0.000008 | 0.000000 | 0.000000 | 0.000005 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
1.3333 | 2.00 | 4.00 | 1.3333 | 2.00 | 4.00 | |
1.5309 | 11.33 | 148.00 | 1.1852 | 9.00 | 120.00 |
Geometric batch size |
Arbitrary batch size |
|||||
1 | 0.869565 | 0.769231 | 0.689655 | 0.864865 | 0.761905 | 0.680851 |
2 | 0.059176 | 0.08193 | 0.088565 | 0.040432 | 0.055286 | 0.059178 |
3 | 0.028637 | 0.042662 | 0.047178 | 0.056233 | 0.071207 | 0.070724 |
4 | 0.015533 | 0.025829 | 0.029730 | 0.012702 | 0.024379 | 0.028500 |
5 | 0.009138 | 0.017213 | 0.020736 | 0.012009 | 0.023001 | 0.026226 |
6 | 0.005694 | 0.012222 | 0.015438 | 0.004622 | 0.012738 | 0.016531 |
7 | 0.003698 | 0.009070 | 0.012021 | 0.003650 | 0.010773 | 0.014187 |
8 | 0.002478 | 0.006951 | 0.009672 | 0.001825 | 0.007351 | 0.010714 |
9 | 0.001700 | 0.005460 | 0.007977 | 0.001325 | 0.006028 | 0.009135 |
10 | 0.001189 | 0.004373 | 0.006709 | 0.000762 | 0.004546 | 0.007492 |
50 | 0.000000 | 0.000033 | 0.000367 | 0.000000 | 0.000016 | 0.000367 |
100 | 0.000000 | 0.000001 | 0.000072 | 0.000000 | 0.000000 | 0.000062 |
150 | 0.000000 | 0.000000 | 0.000022 | 0.000000 | 0.000000 | 0.000016 |
200 | 0.000000 | 0.000000 | 0.000008 | 0.000000 | 0.000000 | 0.000005 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
1.3333 | 2.00 | 4.00 | 1.3333 | 2.00 | 4.00 | |
1.5309 | 11.33 | 148.00 | 1.1852 | 9.00 | 120.00 |
Geometric batch size |
Arbitrary batch size |
|||||
Exp | Er | GEr | Exp | Er | GEr | |
3 | 0.284754 | 0.249906 | 0.251187 | 0.296296 | 0.262144 | 0.263406 |
4 | 0.153814 | 0.155912 | 0.155949 | 0.138271 | 0.140929 | 0.140925 |
5 | 0.103324 | 0.108806 | 0.108638 | 0.104033 | 0.108104 | 0.107992 |
6 | 0.074714 | 0.080158 | 0.079956 | 0.074246 | 0.079344 | 0.079161 |
7 | 0.056661 | 0.061381 | 0.061192 | 0.057051 | 0.061597 | 0.061419 |
8 | 0.04445 | 0.048385 | 0.048220 | 0.044842 | 0.048768 | 0.048605 |
9 | 0.035768 | 0.039006 | 0.038866 | 0.036193 | 0.039482 | 0.039341 |
10 | 0.029358 | 0.032012 | 0.031895 | 0.029750 | 0.032485 | 0.032365 |
50 | 0.000625 | 0.000544 | 0.000547 | 0.000618 | 0.000532 | 0.000536 |
100 | 0.000038 | 0.000024 | 0.000024 | 0.000036 | 0.000022 | 0.000022 |
120 | 0.000014 | 0.000008 | 0.000008 | 0.000013 | 0.000007 | 0.000007 |
150 | 0.000004 | 0.000002 | 0.000002 | 0.000003 | 0.000001 | 0.000001 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
8.57143 | 8.57143 | 8.57143 | 8.57143 | 8.57143 | 8.57143 | |
97.7843 | 83.0030 | 83.5942 | 96.0350 | 81.2536 | 81.8449 |
Geometric batch size |
Arbitrary batch size |
|||||
Exp | Er | GEr | Exp | Er | GEr | |
3 | 0.284754 | 0.249906 | 0.251187 | 0.296296 | 0.262144 | 0.263406 |
4 | 0.153814 | 0.155912 | 0.155949 | 0.138271 | 0.140929 | 0.140925 |
5 | 0.103324 | 0.108806 | 0.108638 | 0.104033 | 0.108104 | 0.107992 |
6 | 0.074714 | 0.080158 | 0.079956 | 0.074246 | 0.079344 | 0.079161 |
7 | 0.056661 | 0.061381 | 0.061192 | 0.057051 | 0.061597 | 0.061419 |
8 | 0.04445 | 0.048385 | 0.048220 | 0.044842 | 0.048768 | 0.048605 |
9 | 0.035768 | 0.039006 | 0.038866 | 0.036193 | 0.039482 | 0.039341 |
10 | 0.029358 | 0.032012 | 0.031895 | 0.029750 | 0.032485 | 0.032365 |
50 | 0.000625 | 0.000544 | 0.000547 | 0.000618 | 0.000532 | 0.000536 |
100 | 0.000038 | 0.000024 | 0.000024 | 0.000036 | 0.000022 | 0.000022 |
120 | 0.000014 | 0.000008 | 0.000008 | 0.000013 | 0.000007 | 0.000007 |
150 | 0.000004 | 0.000002 | 0.000002 | 0.000003 | 0.000001 | 0.000001 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
8.57143 | 8.57143 | 8.57143 | 8.57143 | 8.57143 | 8.57143 | |
97.7843 | 83.0030 | 83.5942 | 96.0350 | 81.2536 | 81.8449 |
Geometric batch size |
Arbitrary batch size |
|||||
H | Cox | IPP | H | Cox | IPP | |
1 | 0.727986 | 0.715778 | 0.761631 | 0.673993 | 0.662037 | 0.720000 |
2 | 0.112936 | 0.119996 | 0.092479 | 0.101235 | 0.105492 | 0.082253 |
3 | 0.052525 | 0.055582 | 0.044531 | 0.060451 | 0.062843 | 0.050677 |
4 | 0.030003 | 0.031571 | 0.026147 | 0.034555 | 0.036224 | 0.028685 |
5 | 0.019157 | 0.020011 | 0.017118 | 0.023752 | 0.024893 | 0.020004 |
6 | 0.013102 | 0.013572 | 0.011986 | 0.017131 | 0.017951 | 0.014566 |
7 | 0.009387 | 0.009637 | 0.008783 | 0.013022 | 0.013625 | 0.011193 |
8 | 0.006955 | 0.007074 | 0.006651 | 0.010219 | 0.010672 | 0.008873 |
9 | 0.005285 | 0.005324 | 0.005164 | 0.008230 | 0.008575 | 0.007217 |
10 | 0.004097 | 0.004087 | 0.004088 | 0.006760 | 0.007025 | 0.005985 |
50 | 0.000009 | 0.000006 | 0.000020 | 0.000151 | 0.000137 | 0.000190 |
100 | 0.000000 | 0.000000 | 0.000000 | 0.000009 | 0.000007 | 0.000019 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
1.92308 | 1.92308 | 1.92308 | 2.85714 | 2.85714 | 2.85714 | |
7.06645 | 6.4406 | 8.94401 | 33.85932 | 31.03417 | 45.15063 |
Geometric batch size |
Arbitrary batch size |
|||||
H | Cox | IPP | H | Cox | IPP | |
1 | 0.727986 | 0.715778 | 0.761631 | 0.673993 | 0.662037 | 0.720000 |
2 | 0.112936 | 0.119996 | 0.092479 | 0.101235 | 0.105492 | 0.082253 |
3 | 0.052525 | 0.055582 | 0.044531 | 0.060451 | 0.062843 | 0.050677 |
4 | 0.030003 | 0.031571 | 0.026147 | 0.034555 | 0.036224 | 0.028685 |
5 | 0.019157 | 0.020011 | 0.017118 | 0.023752 | 0.024893 | 0.020004 |
6 | 0.013102 | 0.013572 | 0.011986 | 0.017131 | 0.017951 | 0.014566 |
7 | 0.009387 | 0.009637 | 0.008783 | 0.013022 | 0.013625 | 0.011193 |
8 | 0.006955 | 0.007074 | 0.006651 | 0.010219 | 0.010672 | 0.008873 |
9 | 0.005285 | 0.005324 | 0.005164 | 0.008230 | 0.008575 | 0.007217 |
10 | 0.004097 | 0.004087 | 0.004088 | 0.006760 | 0.007025 | 0.005985 |
50 | 0.000009 | 0.000006 | 0.000020 | 0.000151 | 0.000137 | 0.000190 |
100 | 0.000000 | 0.000000 | 0.000000 | 0.000009 | 0.000007 | 0.000019 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
1.92308 | 1.92308 | 1.92308 | 2.85714 | 2.85714 | 2.85714 | |
7.06645 | 6.4406 | 8.94401 | 33.85932 | 31.03417 | 45.15063 |
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