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doi: 10.3934/jimo.2021168
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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

Veena Goswami 1, and  M. L. Chaudhry 2,

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

Received  January 2021 Revised  May 2021 Early access September 2021

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.

Keywords: Number served in busy period, batch arrival, Laplace transform, phase-type, functional equation, Langrange inversion theorem.
Mathematics Subject Classification: Primary: 60K25, 68M20; Secondary: 90B22.
Citation: Veena Goswami, M. L. Chaudhry. Explicit results for the distribution of the number of customers served during a busy period for $M^X/PH/1$ queue. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021168
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. Asmussen and H. Albrecher, Ruin Probabilities, vol. 14, Advanced Series on Statistical Science & Applied Probability, Hackensack, NJ, 2010. doi: 10.1142/9789814282536.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]
[9]

P. C. Consul and F. Famoye, Lagrangian Probability Distributions, Springer, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

F. A. Haight, A distribution analogous to the borel-tanner, Biometrika, 48 (1961), 167-173.  doi: 10.1093/biomet/48.1-2.167.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

L. Kleinrock, Queueing Systems, vol. 1, Wiley, New York, 1975. Google Scholar

[20]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, 1999. doi: 10.1137/1.9780898719734.  Google Scholar

[21] J. Medhi, Stochastic Models in Queueing Theory, Academic Press, Amsterdam, 2003.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

N. U. Prabhu, Queues and Inventories, John Wiley & Sons, 1965.  Google Scholar

[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.  Google Scholar

[27]

W. J. Stewart, Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling, Princeton University Press, 2009.  Google Scholar

[28] L. Takács, Combinatorial Methods in the Theory of Stochastic Processes, Wiley, New York, 1967.   Google Scholar
[29]

J. C. Tanner, A problem of interference between two queues, Biometrika, 40 (1953), 58-69.  doi: 10.1093/biomet/40.1-2.58.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. Asmussen and H. Albrecher, Ruin Probabilities, vol. 14, Advanced Series on Statistical Science & Applied Probability, Hackensack, NJ, 2010. doi: 10.1142/9789814282536.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]
[9]

P. C. Consul and F. Famoye, Lagrangian Probability Distributions, Springer, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

F. A. Haight, A distribution analogous to the borel-tanner, Biometrika, 48 (1961), 167-173.  doi: 10.1093/biomet/48.1-2.167.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

L. Kleinrock, Queueing Systems, vol. 1, Wiley, New York, 1975. Google Scholar

[20]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, 1999. doi: 10.1137/1.9780898719734.  Google Scholar

[21] J. Medhi, Stochastic Models in Queueing Theory, Academic Press, Amsterdam, 2003.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

N. U. Prabhu, Queues and Inventories, John Wiley & Sons, 1965.  Google Scholar

[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.  Google Scholar

[27]

W. J. Stewart, Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling, Princeton University Press, 2009.  Google Scholar

[28] L. Takács, Combinatorial Methods in the Theory of Stochastic Processes, Wiley, New York, 1967.   Google Scholar
[29]

J. C. Tanner, A problem of interference between two queues, Biometrika, 40 (1953), 58-69.  doi: 10.1093/biomet/40.1-2.58.  Google Scholar

Figure 1.  Two Phase type distributions with two phases
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Figure 2.  An exponential service phase
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Figure 3.  Two exponential (Erlang-2) service phases in tandem
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Figure 4.  Phase diagram for the generalized Erlang-2 distribution
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Figure 5.  Two exponential phases in parallel
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Figure 6.  The two phase Coxian distribution
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Figure 7.  Phase diagram for the IPP distribution
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Table .  Exponential distribution
Batch size distribution $ R(z) $ $ P(N_m = n),\; n = m, m+1,\dots $
Geometric $ \frac{\mu(1-qz)}{\lambda+\mu-z(\lambda+q\mu)} $ $ \frac{m}{n}\left(\frac{\mu}{\mu+\lambda}\right)^{n}\sum\limits_{j = 0}^{n-m} {2n-m-j-1 \choose n-m-j} \left(\frac{\lambda+\mu q}{\lambda+\mu}\right)^{n-m-j} {n \choose j}(-q)^j $
Arbitrary $ \frac{\mu}{\lambda+\mu-\lambda(g_1 z+g_2 z^2)} $ $ \frac{m}{n}\left(\frac{\mu}{\mu+\lambda}\right)^{n}\sum\limits_{r = 0}^{\infty} {n+r-1 \choose r} {r \choose n-m-r}\left(\frac{\lambda g_1}{\lambda+\mu}\right)^{r}\left(\frac{g_2}{g_1}\right)^{n-m-r} $
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Batch size distribution $ R(z) $ $ P(N_m = n),\; n = m, m+1,\dots $
Geometric $ \frac{\mu(1-qz)}{\lambda+\mu-z(\lambda+q\mu)} $ $ \frac{m}{n}\left(\frac{\mu}{\mu+\lambda}\right)^{n}\sum\limits_{j = 0}^{n-m} {2n-m-j-1 \choose n-m-j} \left(\frac{\lambda+\mu q}{\lambda+\mu}\right)^{n-m-j} {n \choose j}(-q)^j $
Arbitrary $ \frac{\mu}{\lambda+\mu-\lambda(g_1 z+g_2 z^2)} $ $ \frac{m}{n}\left(\frac{\mu}{\mu+\lambda}\right)^{n}\sum\limits_{r = 0}^{\infty} {n+r-1 \choose r} {r \choose n-m-r}\left(\frac{\lambda g_1}{\lambda+\mu}\right)^{r}\left(\frac{g_2}{g_1}\right)^{n-m-r} $
Table .  Erlang distribution
Batch size distribution $ R(z) $ $ P(N_m = n),\; n = m, m+1,\dots $
Geometric $ \left(\frac{\mu(1-qz)}{\lambda+\mu-z(\lambda+q\mu)}\right)^2 $ $ \begin{array}{c} \frac{m}{n}\left(\frac{\mu}{\mu+\lambda}\right)^{nk}\sum\limits_{j = 0}^{n-m} {nk+n-m-j-1 \choose n-m-j} {nk \choose j} (-q)^j \\ \times \left(\frac{\lambda+\mu q}{\lambda+\mu}\right)^{n-m-j} \end{array} $
Arbitrary $ \left(\frac{\mu}{\lambda+\mu-\lambda(g_1 z+g_2 z^2)}\right)^2 $ $ \begin{array}{c} \frac{m}{n}{\left( {\frac{\mu }{{\mu + \lambda }}} \right)^{nk}}\sum\limits_{r = 0}^\infty {\left( \begin{array}{c} nk + r - 1\\ r \end{array} \right)} \left( \begin{array}{c} r\\ n - m - r \end{array} \right){\left( {\frac{{\lambda {g_1}}}{{\lambda + \mu }}} \right)^r}\\ \times {\left( {\frac{{{g_2}}}{{{g_1}}}} \right)^{n - m - r}} \end{array}$
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Batch size distribution $ R(z) $ $ P(N_m = n),\; n = m, m+1,\dots $
Geometric $ \left(\frac{\mu(1-qz)}{\lambda+\mu-z(\lambda+q\mu)}\right)^2 $ $ \begin{array}{c} \frac{m}{n}\left(\frac{\mu}{\mu+\lambda}\right)^{nk}\sum\limits_{j = 0}^{n-m} {nk+n-m-j-1 \choose n-m-j} {nk \choose j} (-q)^j \\ \times \left(\frac{\lambda+\mu q}{\lambda+\mu}\right)^{n-m-j} \end{array} $
Arbitrary $ \left(\frac{\mu}{\lambda+\mu-\lambda(g_1 z+g_2 z^2)}\right)^2 $ $ \begin{array}{c} \frac{m}{n}{\left( {\frac{\mu }{{\mu + \lambda }}} \right)^{nk}}\sum\limits_{r = 0}^\infty {\left( \begin{array}{c} nk + r - 1\\ r \end{array} \right)} \left( \begin{array}{c} r\\ n - m - r \end{array} \right){\left( {\frac{{\lambda {g_1}}}{{\lambda + \mu }}} \right)^r}\\ \times {\left( {\frac{{{g_2}}}{{{g_1}}}} \right)^{n - m - r}} \end{array}$
Table .  Generalized Erlang distribution
Batch size distribution $ R(z) $ $ P(N_m = n),\; n = m, m+1,\dots $
Geometric $ \frac{\mu_1\mu_2}{\lambda+\mu_1-z(\lambda+q\mu_1)} \\ \times \frac{(1-qz)^2}{\lambda+\mu_2-z(\lambda+q\mu_2)} $ $ \begin{array}{c} \frac{m}{n}\prod\limits_{i = 1}^2 {{{\left( {\frac{{{\mu _i}}}{{{\mu _i} + \lambda }}} \right)}^n}} \sum\limits_{i = 0}^{n - m} {\left( \begin{array}{c} 2n\\ i \end{array} \right)} {( - q)^i}\sum\limits_{j = 0}^{n - m - i} {\left( \begin{array}{c} n + j - 1\\ j \end{array} \right)} \\ \times {\left( {\frac{{\lambda + q{\mu _2}}}{{\lambda + {\mu _2}}}} \right)^j}\left( \begin{array}{c} 2n - m - i - j - 1\\ n - m - i - j \end{array} \right){\left( {\frac{{\lambda + q{\mu _1}}}{{\lambda + {\mu _1}}}} \right)^{n - m - i - j}} \end{array} $
Arbitrary $ \begin{array}{c} \prod\limits_{i = 1}^{2}\frac{\mu_i}{\mu_i+\lambda} \\ \times \prod\limits_{i = 1}^{4}(1-z/\omega_i)^{-1} \end{array} $ $ \begin{array}{c} \frac{m}{n}\prod\limits_{i = 1}^{2}\left(\frac{\mu_i}{\mu_i+\lambda}\right)^n\sum\limits_{i = 0}^{n-m}{n+i-1 \choose i}\omega_1^{-i}\sum\limits_{j = 0}^{n-m-i}{n+j-1 \choose j} \\ \times \omega_2^{-j}\sum\limits_{k = 0}^{n-m-i-j}{n+k-1 \choose k}\omega_3^{-k} {2n-m-i-j-k-1 \choose n-m-i-j-k} \\ \times\omega_4^{i+j+k+m-n} \end{array}$
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Batch size distribution $ R(z) $ $ P(N_m = n),\; n = m, m+1,\dots $
Geometric $ \frac{\mu_1\mu_2}{\lambda+\mu_1-z(\lambda+q\mu_1)} \\ \times \frac{(1-qz)^2}{\lambda+\mu_2-z(\lambda+q\mu_2)} $ $ \begin{array}{c} \frac{m}{n}\prod\limits_{i = 1}^2 {{{\left( {\frac{{{\mu _i}}}{{{\mu _i} + \lambda }}} \right)}^n}} \sum\limits_{i = 0}^{n - m} {\left( \begin{array}{c} 2n\\ i \end{array} \right)} {( - q)^i}\sum\limits_{j = 0}^{n - m - i} {\left( \begin{array}{c} n + j - 1\\ j \end{array} \right)} \\ \times {\left( {\frac{{\lambda + q{\mu _2}}}{{\lambda + {\mu _2}}}} \right)^j}\left( \begin{array}{c} 2n - m - i - j - 1\\ n - m - i - j \end{array} \right){\left( {\frac{{\lambda + q{\mu _1}}}{{\lambda + {\mu _1}}}} \right)^{n - m - i - j}} \end{array} $
Arbitrary $ \begin{array}{c} \prod\limits_{i = 1}^{2}\frac{\mu_i}{\mu_i+\lambda} \\ \times \prod\limits_{i = 1}^{4}(1-z/\omega_i)^{-1} \end{array} $ $ \begin{array}{c} \frac{m}{n}\prod\limits_{i = 1}^{2}\left(\frac{\mu_i}{\mu_i+\lambda}\right)^n\sum\limits_{i = 0}^{n-m}{n+i-1 \choose i}\omega_1^{-i}\sum\limits_{j = 0}^{n-m-i}{n+j-1 \choose j} \\ \times \omega_2^{-j}\sum\limits_{k = 0}^{n-m-i-j}{n+k-1 \choose k}\omega_3^{-k} {2n-m-i-j-k-1 \choose n-m-i-j-k} \\ \times\omega_4^{i+j+k+m-n} \end{array}$
Table .  Hyperexponential distribution
Batch size distribution $ R(z) $ $ P(N_m = n),\; n = m, m+1,\dots $
Geometric $ \begin{array}{c} \frac{{(\lambda {D_1} + {D_2})\left[ {1 - z\frac{{\lambda {D_1} + q{D_2}}}{{\lambda {D_1} + {D_2}}}} \right]}}{{\lambda + {\mu _1} - z(\lambda + q{\mu _1})}}\\ \times \frac{{(1 - qz)}}{{\lambda + {\mu _2} - z(\lambda + q{\mu _2})}} \end{array}$ $ \begin{array}{c} \frac{m}{n}\left(\frac{\lambda D_1+D_2}{\prod\limits_{i = 1}^{2}(\lambda+\mu_i)}\right)^n \sum\limits_{i = 0}^{n-m} {n \choose i}\left(\frac{\lambda D_1+q D_2}{\lambda D_1+D_2}\right)^i \sum\limits_{j = 0}^{n-m-i} {n \choose j} \\ \times (-q)^j \sum\limits_{k = 0}^{n-m-i-j}{n+k-1 \choose k}\left(\frac{\lambda+q \mu_1 }{\lambda+\mu_1}\right)^{k} \\ \times {2n-m-i-j-k-1 \choose n-m-i-j-k} \left(\frac{\lambda+q \mu_2 }{\lambda+\mu_2}\right)^{n-m-i-j-k} \end{array} $
Arbitrary $ \frac{(\lambda D_1+D_2)\prod\limits_{i = 1}^{2}(1-z/\xi_i)}{\prod\limits_{i = 1}^{2}(\lambda+\mu_i) \prod\limits_{i = 1}^{4}(1-z/\omega_i)} $ $ \begin{array}{c} \frac{m}{n}\left(\frac{\lambda D_1+D_2}{\prod\limits_{i = 1}^{2}(\lambda+\mu_i)}\right)^n \sum\limits_{i = 0}^{n-m}{n \choose i}(-\xi_1)^{-i}\sum\limits_{j = 0}^{n-m-i}{n \choose j}(-\xi_2)^{-j} \\ \times \sum\limits_{k_1 = 0}^{n-m-i-j}{n+k_1-1 \choose k_1} \omega_1^{-k_1} \sum\limits_{k_2 = 0}^{n-m-i-j-k_1} {n+k_2-1 \choose k_2} \\ \times \omega_2^{-k_2}\sum\limits_{k_3 = 0}^{n-m-i-j-k_1-k_2} {n+k_3-1 \choose k_3}\omega_3^{-k_3} \\ \times \ {2n-m-i-j-k_1-k_2-k_3-1 \choose n-m-i-j-k_1-k_2-k_3} \omega_4^{i+j+k_1+k_2+k_3+m-n} \end{array} $
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Batch size distribution $ R(z) $ $ P(N_m = n),\; n = m, m+1,\dots $
Geometric $ \begin{array}{c} \frac{{(\lambda {D_1} + {D_2})\left[ {1 - z\frac{{\lambda {D_1} + q{D_2}}}{{\lambda {D_1} + {D_2}}}} \right]}}{{\lambda + {\mu _1} - z(\lambda + q{\mu _1})}}\\ \times \frac{{(1 - qz)}}{{\lambda + {\mu _2} - z(\lambda + q{\mu _2})}} \end{array}$ $ \begin{array}{c} \frac{m}{n}\left(\frac{\lambda D_1+D_2}{\prod\limits_{i = 1}^{2}(\lambda+\mu_i)}\right)^n \sum\limits_{i = 0}^{n-m} {n \choose i}\left(\frac{\lambda D_1+q D_2}{\lambda D_1+D_2}\right)^i \sum\limits_{j = 0}^{n-m-i} {n \choose j} \\ \times (-q)^j \sum\limits_{k = 0}^{n-m-i-j}{n+k-1 \choose k}\left(\frac{\lambda+q \mu_1 }{\lambda+\mu_1}\right)^{k} \\ \times {2n-m-i-j-k-1 \choose n-m-i-j-k} \left(\frac{\lambda+q \mu_2 }{\lambda+\mu_2}\right)^{n-m-i-j-k} \end{array} $
Arbitrary $ \frac{(\lambda D_1+D_2)\prod\limits_{i = 1}^{2}(1-z/\xi_i)}{\prod\limits_{i = 1}^{2}(\lambda+\mu_i) \prod\limits_{i = 1}^{4}(1-z/\omega_i)} $ $ \begin{array}{c} \frac{m}{n}\left(\frac{\lambda D_1+D_2}{\prod\limits_{i = 1}^{2}(\lambda+\mu_i)}\right)^n \sum\limits_{i = 0}^{n-m}{n \choose i}(-\xi_1)^{-i}\sum\limits_{j = 0}^{n-m-i}{n \choose j}(-\xi_2)^{-j} \\ \times \sum\limits_{k_1 = 0}^{n-m-i-j}{n+k_1-1 \choose k_1} \omega_1^{-k_1} \sum\limits_{k_2 = 0}^{n-m-i-j-k_1} {n+k_2-1 \choose k_2} \\ \times \omega_2^{-k_2}\sum\limits_{k_3 = 0}^{n-m-i-j-k_1-k_2} {n+k_3-1 \choose k_3}\omega_3^{-k_3} \\ \times \ {2n-m-i-j-k_1-k_2-k_3-1 \choose n-m-i-j-k_1-k_2-k_3} \omega_4^{i+j+k_1+k_2+k_3+m-n} \end{array} $
Table .  Cox-$ C_2 $ distribution
Batch size distribution $ R(z) $ $ P(N_m = n),\; n = m, m+1,\dots $
Geometric $\begin{array}{c} \frac{{(\lambda {D_1} + {D_2})\left[ {1 - z\frac{{\lambda {D_1} + q{D_2}}}{{\lambda {D_1} + {D_2}}}} \right]}}{{\lambda + {\mu _1} - z(\lambda + q{\mu _1})}}\\ \times \frac{{1 - qz}}{{\lambda + {\mu _2} - z(\lambda + q{\mu _2})}} \end{array}$ $ \begin{array}{c} \frac{m}{n}\left(\frac{\lambda D_1+D_2}{\prod\limits_{i = 1}^{2}(\mu_i+\lambda)}\right)^{n} \sum\limits_{i = 0}^{n-m}{n \choose i}(-1)^i \left(\frac{\lambda(1-\beta)+q \mu_2}{\lambda(1-\beta)+\mu_2}\right)^{i} \\ \times \sum\limits_{j = 0}^{n-m-i} {n \choose j}(-q)^j \sum\limits_{k = 0}^{n-m-i-j}{n+k-1 \choose k}\left(\frac{\lambda+q \mu_1 }{\lambda+\mu_1}\right)^k \\ \times {2n-m-i-j-k-1 \choose n-m-i-j-k}\left(\frac{\lambda+q \mu_2 }{\lambda+\mu_2}\right)^{n-m-i-j-k} \end{array} $
Arbitrary $ \frac{(\lambda D_1+D_2)\prod\limits_{i = 1}^{2}(1-z/\xi_i)}{\prod\limits_{i = 1}^{2}(\lambda+\mu_i) \prod\limits_{i = 1}^{4}(1-z/\omega_i)} $ $ \begin{array}{c} \frac{1}{n}\left(\frac{\lambda D_1+D_2}{\prod\limits_{i = 1}^{2}(\lambda+\mu_i) }\right)^n \sum\limits_{i = 0}^{n-1}{n \choose i}(-\xi_1)^{-i}\sum\limits_{j = 0}^{n-1-j}{n \choose j}(-\xi_2)^{-j} \\ \times\sum\limits_{k = 0}^{n-1-i-j}{n+k-1 \choose k} \omega_1^{-k} \sum\limits_{l = 0}^{n-1-i-j-k} {n+l-1 \choose l}\omega_2^{-l} \\ \times \sum\limits_{m = 0}^{n-1-i-j-k-l} {n+m-1 \choose m}\omega_3^{-m} \\ {2n-i-j-k-l-m-2 \choose n-i-j-k-l-m-1} \omega_4^{i+j+k+l+m+1-n} \end{array} $
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Batch size distribution $ R(z) $ $ P(N_m = n),\; n = m, m+1,\dots $
Geometric $\begin{array}{c} \frac{{(\lambda {D_1} + {D_2})\left[ {1 - z\frac{{\lambda {D_1} + q{D_2}}}{{\lambda {D_1} + {D_2}}}} \right]}}{{\lambda + {\mu _1} - z(\lambda + q{\mu _1})}}\\ \times \frac{{1 - qz}}{{\lambda + {\mu _2} - z(\lambda + q{\mu _2})}} \end{array}$ $ \begin{array}{c} \frac{m}{n}\left(\frac{\lambda D_1+D_2}{\prod\limits_{i = 1}^{2}(\mu_i+\lambda)}\right)^{n} \sum\limits_{i = 0}^{n-m}{n \choose i}(-1)^i \left(\frac{\lambda(1-\beta)+q \mu_2}{\lambda(1-\beta)+\mu_2}\right)^{i} \\ \times \sum\limits_{j = 0}^{n-m-i} {n \choose j}(-q)^j \sum\limits_{k = 0}^{n-m-i-j}{n+k-1 \choose k}\left(\frac{\lambda+q \mu_1 }{\lambda+\mu_1}\right)^k \\ \times {2n-m-i-j-k-1 \choose n-m-i-j-k}\left(\frac{\lambda+q \mu_2 }{\lambda+\mu_2}\right)^{n-m-i-j-k} \end{array} $
Arbitrary $ \frac{(\lambda D_1+D_2)\prod\limits_{i = 1}^{2}(1-z/\xi_i)}{\prod\limits_{i = 1}^{2}(\lambda+\mu_i) \prod\limits_{i = 1}^{4}(1-z/\omega_i)} $ $ \begin{array}{c} \frac{1}{n}\left(\frac{\lambda D_1+D_2}{\prod\limits_{i = 1}^{2}(\lambda+\mu_i) }\right)^n \sum\limits_{i = 0}^{n-1}{n \choose i}(-\xi_1)^{-i}\sum\limits_{j = 0}^{n-1-j}{n \choose j}(-\xi_2)^{-j} \\ \times\sum\limits_{k = 0}^{n-1-i-j}{n+k-1 \choose k} \omega_1^{-k} \sum\limits_{l = 0}^{n-1-i-j-k} {n+l-1 \choose l}\omega_2^{-l} \\ \times \sum\limits_{m = 0}^{n-1-i-j-k-l} {n+m-1 \choose m}\omega_3^{-m} \\ {2n-i-j-k-l-m-2 \choose n-i-j-k-l-m-1} \omega_4^{i+j+k+l+m+1-n} \end{array} $
Table .  IPP distribution
Batch size distribution $ R(z) $ $ P(N_m = n),\; n = m, m+1,\dots $
Geometric $ \begin{array}{c} \frac{{(\lambda {D_1} + {D_2})\left[ {1 - z\frac{{\lambda {D_1} + q{D_2}}}{{\lambda {D_1} + {D_2}}}} \right]}}{{\lambda + {\mu _1} - z(\lambda + q{\mu _1})}}\\ \times \frac{{1 - qz}}{{\lambda + {\mu _2} - z(\lambda + q{\mu _2})}} \end{array}$ $ \begin{array}{c} \frac{m}{n}\left(\frac{\mu(\lambda+r_2)}{u_3}\right)^n \sum\limits_{i = 0}^{n-m}{n+i-1 \choose i}\psi_1^{-i} \sum\limits_{j = 0}^{n-m-i} {n \choose j} \\ \times \left(- \frac{\lambda+q r_2 }{\lambda+r_2}\right)^j \sum\limits_{k = 0}^{n-m-i-j}{n \choose k}(-q)^{k} \\ \times {2n-m-i-j-k-1 \choose n-m-i-j-k}\psi_2^{i+j+k+m-n} \end{array} $
Arbitrary $ \frac{\lambda D_1+D_2-\lambda D_1(g_1 z+g_2 z^2)}{(\lambda^2 g_2^2 \prod\limits_{i = 1}^{4} \omega_i)\prod\limits_{i = 1}^{4}(1-z/\omega_i)} $ $ \begin{array}{c} \frac{m}{n}\left(\frac{\mu(\lambda+r_2)}{u_3}\right)^n \sum\limits_{i = 0}^{n-m}{n \choose i}(-\xi_1)^{-i}\sum\limits_{j = 0}^{n-m-i}{n \choose j}(-\xi_2)^{-j} \\ \times \sum\limits_{k_1 = 0}^{n-m-i-j}{n+k_1-1 \choose k_1}\omega_1^{-k_1}\sum\limits_{k_2 = 0}^{n-m-i-j-k_1} {n+k_2-1 \choose k_2} \\ \times \omega_2^{-k_2} \sum\limits_{k_3 = 0}^{n-m-i-j-k_1-k_2} {n+k_3-1 \choose k_3}\omega_3^{-k_3} \\ \times {2n-m-i-j-k_1-k_2-k_3-1 \choose n-m-i-j-k_1-k_2-k_3} \omega_4^{i+j+k_1+l+k_2+k_3+m-n} \end{array}$
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Batch size distribution $ R(z) $ $ P(N_m = n),\; n = m, m+1,\dots $
Geometric $ \begin{array}{c} \frac{{(\lambda {D_1} + {D_2})\left[ {1 - z\frac{{\lambda {D_1} + q{D_2}}}{{\lambda {D_1} + {D_2}}}} \right]}}{{\lambda + {\mu _1} - z(\lambda + q{\mu _1})}}\\ \times \frac{{1 - qz}}{{\lambda + {\mu _2} - z(\lambda + q{\mu _2})}} \end{array}$ $ \begin{array}{c} \frac{m}{n}\left(\frac{\mu(\lambda+r_2)}{u_3}\right)^n \sum\limits_{i = 0}^{n-m}{n+i-1 \choose i}\psi_1^{-i} \sum\limits_{j = 0}^{n-m-i} {n \choose j} \\ \times \left(- \frac{\lambda+q r_2 }{\lambda+r_2}\right)^j \sum\limits_{k = 0}^{n-m-i-j}{n \choose k}(-q)^{k} \\ \times {2n-m-i-j-k-1 \choose n-m-i-j-k}\psi_2^{i+j+k+m-n} \end{array} $
Arbitrary $ \frac{\lambda D_1+D_2-\lambda D_1(g_1 z+g_2 z^2)}{(\lambda^2 g_2^2 \prod\limits_{i = 1}^{4} \omega_i)\prod\limits_{i = 1}^{4}(1-z/\omega_i)} $ $ \begin{array}{c} \frac{m}{n}\left(\frac{\mu(\lambda+r_2)}{u_3}\right)^n \sum\limits_{i = 0}^{n-m}{n \choose i}(-\xi_1)^{-i}\sum\limits_{j = 0}^{n-m-i}{n \choose j}(-\xi_2)^{-j} \\ \times \sum\limits_{k_1 = 0}^{n-m-i-j}{n+k_1-1 \choose k_1}\omega_1^{-k_1}\sum\limits_{k_2 = 0}^{n-m-i-j-k_1} {n+k_2-1 \choose k_2} \\ \times \omega_2^{-k_2} \sum\limits_{k_3 = 0}^{n-m-i-j-k_1-k_2} {n+k_3-1 \choose k_3}\omega_3^{-k_3} \\ \times {2n-m-i-j-k_1-k_2-k_3-1 \choose n-m-i-j-k_1-k_2-k_3} \omega_4^{i+j+k_1+l+k_2+k_3+m-n} \end{array}$
Table 1.  PH service-time distributions
$ \mu_1=3 $, $ \mu_2=1 $, $ r_{10}=1/3 $, $ r_{12}=2/3 $, $ r_{20}=1 $, $ r_{21}=0 $
Geometric batch size
$ p=0.6 $, $ q=0.4 $		 Arbitrary batch size
$ g_1=0.4 $, $ g_2=0.6 $
$ n $ $ \rho=0.25 $ $ \rho=0.5 $ $ \rho=0.75 $ $ \rho=0.25 $ $ \rho=0.5 $ $ \rho=0.75 $
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
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$ E(N) $ 1.3333 2.00 4.00 1.3333 2.00 4.00
$ V(N) $ 1.5309 11.33 148.00 1.1852 9.00 120.00
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$ \mu_1=3 $, $ \mu_2=1 $, $ r_{10}=1/3 $, $ r_{12}=2/3 $, $ r_{20}=1 $, $ r_{21}=0 $
Geometric batch size
$ p=0.6 $, $ q=0.4 $		 Arbitrary batch size
$ g_1=0.4 $, $ g_2=0.6 $
$ n $ $ \rho=0.25 $ $ \rho=0.5 $ $ \rho=0.75 $ $ \rho=0.25 $ $ \rho=0.5 $ $ \rho=0.75 $
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
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$ E(N) $ 1.3333 2.00 4.00 1.3333 2.00 4.00
$ V(N) $ 1.5309 11.33 148.00 1.1852 9.00 120.00
Table 2.  Exponential (Exp), Erlang (Er), Generalized-Erlang (GEr) service-time distributions
Geometric batch size
$ p=0.8 $, $ q=0.2 $, $ m=3 $, $ \rho=0.65 $		 Arbitrary batch size
$ g_1=0.7 $, $ g_2=0.3 $, $ m=3 $, $ \rho=0.65 $
$ n $ 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
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$ E(N) $ 8.57143 8.57143 8.57143 8.57143 8.57143 8.57143
$ V(N) $ 97.7843 83.0030 83.5942 96.0350 81.2536 81.8449
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Geometric batch size
$ p=0.8 $, $ q=0.2 $, $ m=3 $, $ \rho=0.65 $		 Arbitrary batch size
$ g_1=0.7 $, $ g_2=0.3 $, $ m=3 $, $ \rho=0.65 $
$ n $ 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
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$ E(N) $ 8.57143 8.57143 8.57143 8.57143 8.57143 8.57143
$ V(N) $ 97.7843 83.0030 83.5942 96.0350 81.2536 81.8449
Table 3.  Hyperexponential (H), Coxian (Cox), IPP service-time distributions
Geometric batch size
$ p=0.8 $, $ q=0.2 $, $ \rho=0.48 $		 Arbitrary batch size
$ g_1=0.7 $, $ g_2=0.3 $, $ \rho=0.65 $
$ n $ 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
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$ E(N) $ 1.92308 1.92308 1.92308 2.85714 2.85714 2.85714
$ V(N) $ 7.06645 6.4406 8.94401 33.85932 31.03417 45.15063
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Geometric batch size
$ p=0.8 $, $ q=0.2 $, $ \rho=0.48 $		 Arbitrary batch size
$ g_1=0.7 $, $ g_2=0.3 $, $ \rho=0.65 $
$ n $ 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
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$ E(N) $ 1.92308 1.92308 1.92308 2.85714 2.85714 2.85714
$ V(N) $ 7.06645 6.4406 8.94401 33.85932 31.03417 45.15063
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