\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Abstract Full Text(HTML) Figure(7) / Table(9) Related Papers Cited by
  • 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.

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

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Two Phase type distributions with two phases

    Figure 2.  An exponential service phase

    Figure 3.  Two exponential (Erlang-2) service phases in tandem

    Figure 4.  Phase diagram for the generalized Erlang-2 distribution

    Figure 5.  Two exponential phases in parallel

    Figure 6.  The two phase Coxian distribution

    Figure 7.  Phase diagram for the IPP distribution

    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} $
     | Show Table
    DownLoad: CSV

    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}$
     | Show Table
    DownLoad: CSV

    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}$
     | Show Table
    DownLoad: CSV

    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} $
     | Show Table
    DownLoad: CSV

    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} $
     | Show Table
    DownLoad: CSV

    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}$
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
  • [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. MedhiStochastic 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. TakácsCombinatorial 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.
  • 加载中

Figures(7)

Tables(9)

SHARE

Article Metrics

HTML views(422) PDF downloads(445) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return