Explicit results for the distribution of the number of customers served during a busy period for $M^X/PH/1$ queue

Veena Goswami; M. L. Chaudhry

doi:10.3934/jimo.2021168

Article Contents

2022, Volume 18, Issue 6: 4471-4490. Doi: 10.3934/jimo.2021168

This issue Previous Article Equity-based incentive to coordinate shareholder-manager interests under information asymmetry Next Article Multi-objective chance-constrained blending optimization of zinc smelter under stochastic uncertainty

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: December 31, 2020

Revised: April 30, 2021

Early access: September 2021

Published: September 2022

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Abstract

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.

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Mathematics Subject Classification: Primary: 60K25, 68M20; Secondary: 90B22.

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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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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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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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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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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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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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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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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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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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