doi: 10.3934/jimo.2021078
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Steady-state and first passage time distributions for waiting times in the $ MAP/M/s+G $ queueing model with generally distributed patience times

1. 

Electrical and Electronics Eng. Dept., Bilkent University, Bilkent 06800, Ankara, Turkey

2. 

University of Tabriz, Tabriz, East Azerbaijan, Iran

* Corresponding author: Nail Akar

Received  August 2020 Revised  December 2020 Early access April 2021

We study the $ MAP/M/s+G $ queueing model that arises in various multi-server engineering problems including telephone call centers, under the assumption of MAP (Markovian Arrival Process) arrivals, exponentially distributed service times, infinite waiting room, and generally distributed patience times. Using sample-path arguments, we propose to obtain the steady-state distribution of the virtual waiting time and subsequently the other relevant performance metrics of interest via the steady-state solution of a certain Continuous Feedback Fluid Queue (CFFQ). The proposed method is exact when the patience time is a discrete random variable and is asymptotically exact when it is continuous/hybrid, for which case discretization of the patience time distribution is required giving rise to a computational complexity depending linearly on the number of discretization levels. Additionally, a novel method is proposed to accurately obtain the first passage time distributions for the virtual and actual waiting times again using CFFQs while approximating the deterministic time horizons by Erlang distributions or more efficient Concentrated Matrix Exponential (CME) distributions. Numerical results are presented to validate the effectiveness of the proposed numerical method.

Citation: Omer Gursoy, Kamal Adli Mehr, Nail Akar. Steady-state and first passage time distributions for waiting times in the $ MAP/M/s+G $ queueing model with generally distributed patience times. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021078
References:
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N. Akar, O. Gursoy, G. Horvath and M. Telek, Transient and First Passage Time Distributions of First- and Second-Order Multi-Regime Markov Fluid Queues via ME-fication, accepted for publication in Methodology and Computing in Applied Probability, 2020. Google Scholar

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N. Akar and K. Sohraby, Infinite- and finite-buffer Markov fluid queues: A unified analysis, J. Appl. Probab., 41 (2004), 557-569.  doi: 10.1239/jap/1082999086.  Google Scholar

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Q.-M. He and H. Wu, Multi-layer MMFF processes and the MAP/PH/K+GI queue: Theory and algorithms, Queueing Models and Service Management, 3 (2020), 37-87.   Google Scholar

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[31]

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H. E. Kankaya and N. Akar, Solving multi-regime feedback fluid queues, Stochastic Models, 24 (2008), 425-450.  doi: 10.1080/15326340802232285.  Google Scholar

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K. Kawanishi and T. Takine, MAP/M/c and M/PH/c queues with constant impatience times, Queueing Systems, 82 (2016), 381-420.  doi: 10.1007/s11134-015-9455-9.  Google Scholar

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M. MandjesD. Mitra and W. Scheinhardt, Models of network access using feedback fluid queues, Queueing Syst., 44 (2003), 365-398.  doi: 10.1023/A:1025147422141.  Google Scholar

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show all references

References:
[1]

S. AhnA. L. Badescu and V. Ramaswami, Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier, Queueing Syst., 55 (2007), 207-222.  doi: 10.1007/s11134-007-9017-x.  Google Scholar

[2]

S. Ahn and V. Ramaswami, Efficient algorithms for transient analysis of stochastic fluid flow models, Journal of Applied Probability, 42 (2005), 531-549.  doi: 10.1239/jap/1118777186.  Google Scholar

[3]

N. Akar, O. Gursoy, G. Horvath and M. Telek, Transient and First Passage Time Distributions of First- and Second-Order Multi-Regime Markov Fluid Queues via ME-fication, accepted for publication in Methodology and Computing in Applied Probability, 2020. Google Scholar

[4]

N. Akar and K. Sohraby, Infinite- and finite-buffer Markov fluid queues: A unified analysis, J. Appl. Probab., 41 (2004), 557-569.  doi: 10.1239/jap/1082999086.  Google Scholar

[5]

Z. AksinM. Armony and V. Mehrotra, The modern call center: A multi-disciplinary perspective on operations management research, Production and Operations Management, 16 (2007), 665-688.  doi: 10.1111/j.1937-5956.2007.tb00288.x.  Google Scholar

[6]

A. T. Andersen and B. F. Nielsen, A Markovian approach for modeling packet traffic with long-range dependence, IEEE Journal on Selected Areas in Communications, 16 (1998), 719-732.  doi: 10.1109/49.700908.  Google Scholar

[7]

D. AnickD. Mitra and M. M. Sondhi, Stochastic theory of a data-handling system with multiple sources, Bell System Technical Journal, 61 (1982), 1871-1894.  doi: 10.1002/j.1538-7305.1982.tb03089.x.  Google Scholar

[8]

S. Asmussen and M. Bladt, Renewal theory and queueing algorithms for matrix-exponential distributions, in Matrix-Analytic Methods in Stochastic Models, Lecture Notes in Pure and Appl. Math., 183, Dekker, New York, 1997,313–341.  Google Scholar

[9]

S. Asmussen and G. Koole, Marked point processes as limits of Markovian arrival streams, Journal of Applied Probability, 30 (1993), 365-372.  doi: 10.2307/3214845.  Google Scholar

[10]

S. Asmussen and J. R. Møller, Calculation of the steady state waiting time distribution in GI/PH/c and MAP/PH/c queues, Queueing Systems, 37 (2001), 9-29.  doi: 10.1023/A:1011083915877.  Google Scholar

[11]

F. Baccelli and G. Hebuterne, On queues with impatient customers, in Performance '81, (Amsterdam, 1981), North-Holland, Amsterdam-New York, 1981., 159–179.  Google Scholar

[12]

A. BadescuS. Drekic and D. Landriault, On the analysis of a multi-threshold Markovian risk model, Scandinavian Actuarial Journal, 2007 (2007), 248-260.  doi: 10.1080/03461230701554080.  Google Scholar

[13]

N. G. Bean and B. F. Nielsen, Quasi-birth-and-death processes with rational arrival process components, Stochastic Models, 26 (2010), 309-334.  doi: 10.1080/15326349.2010.498311.  Google Scholar

[14]

N. G. Bean and M. M. O'Reilly, Performance measures of a multi-layer Markovian fluid model, Annals of Operations Research, 160 (2008), 99-120.  doi: 10.1007/s10479-007-0299-5.  Google Scholar

[15]

A. Brandt and M. Brandt, On the M(n)/M(n)/s queue with impatient calls, Performance Evaluation, 35 (1999), 1-18.   Google Scholar

[16]

A. Brandt and M. Brandt, Asymptotic results and a Markovian approximation for the M(n)/M(n)/s+GI system, Queueing Systems, 41 (2002), 73-94.  doi: 10.1023/A:1015781818360.  Google Scholar

[17]

L. BrownN. GansA. MandelbaumA. SakovH. ShenS. Zeltyn and L. Zhao, Statistical analysis of a telephone call center, Journal of the American Statistical Association, 100 (2005), 36-50.  doi: 10.1198/016214504000001808.  Google Scholar

[18]

P. Buchholz and M. Telek, Stochastic Petri nets with matrix exponentially distributed firing times, Performance Evaluation, 67 (2010), 1373-1385.  doi: 10.1016/j.peva.2010.08.023.  Google Scholar

[19]

P. Buchholz and M. Telek, Rational processes related to communicating Markov processes, Journal of Applied Probability, 49 (2012), 40-59.  doi: 10.1239/jap/1331216833.  Google Scholar

[20]

G. Casale, Building accurate workload models using Markovian arrival processes, in Proceedings of the ACM SIGMETRICS Joint International Conference on Measurement and Modeling of Computer Systems, SIGMETRICS '11, ACM, New York, NY, USA, (2011), 357–358. doi: 10.1145/1993744.1993783.  Google Scholar

[21]

G. CasaleE. Z. Zhang and E. Smirni, KPC-Toolbox: Best recipes for automatic trace fitting using Markovian Arrival Processes, Performance Evaluation, 67 (2010), 873-896.  doi: 10.1016/j.peva.2009.12.003.  Google Scholar

[22]

B. D. ChoiB. Kim and D. Zhu, MAP/M/c Queue with Constant Impatient Time, Mathematics of Operations Research, 29 (2004), 309-325.  doi: 10.1287/moor.1030.0081.  Google Scholar

[23]

A. da Silva Soares and G. Latouche, Fluid queues with level dependent evolution, European Journal of Operational Research, 196 (2009), 1041-1048.  doi: 10.1016/j.ejor.2008.05.010.  Google Scholar

[24]

M. W. Fackrell, Characterization of Matrix-Exponential Distributions, PhD thesis, The University of Adelaide, 2003. Google Scholar

[25]

N. GansG. Koole and A. Mandelbaum, Telephone call centers: Tutorial, review, and research prospects, Manufacturing & Service Operations Management, 5 (2003), 79-141.  doi: 10.1287/msom.5.2.79.16071.  Google Scholar

[26]

O. GarnettA. Mandelbaum and M. Reiman, Designing a call center with impatient customers, Manufacturing & Service Operations Management, 4 (2002), 208-227.  doi: 10.1287/msom.4.3.208.7753.  Google Scholar

[27]

Q.-M. He and H. Wu, Multi-layer MMFF processes and the MAP/PH/K+GI queue: Theory and algorithms, Queueing Models and Service Management, 3 (2020), 37-87.   Google Scholar

[28]

Q.-M. He and H. Zhang, On matrix exponential distributions, Advances in Applied Probability, 39 (2007), 271-292.  doi: 10.1239/aap/1175266478.  Google Scholar

[29]

G. HorváthI. Horváth and M. Telek, High order concentrated matrix-exponential distributions, Stochastic Models, 36 (2020), 176-192.  doi: 10.1080/15326349.2019.1702058.  Google Scholar

[30]

I. Horváth, O. Sáfár, M. Telek and B. Zámbó, Concentrated matrix exponential distributions, in Computer Performance Engineering (eds. D. Fiems, M. Paolieri and A. N. Platis), Springer International Publishing, Cham, (2016), 18–31. Google Scholar

[31]

O. JouiniG. Koole and A. Roubos, Performance indicators for call centers with impatient customers, IIE Transactions, 45 (2013), 341-354.  doi: 10.1080/0740817X.2012.712241.  Google Scholar

[32]

H. E. Kankaya and N. Akar, Solving multi-regime feedback fluid queues, Stochastic Models, 24 (2008), 425-450.  doi: 10.1080/15326340802232285.  Google Scholar

[33]

K. Kawanishi and T. Takine, MAP/M/c and M/PH/c queues with constant impatience times, Queueing Systems, 82 (2016), 381-420.  doi: 10.1007/s11134-015-9455-9.  Google Scholar

[34]

C. KimS. DudinO. Taramin and J. Baek, Queueing system $MAP|PH|N|N+R$ with impatient heterogeneous customers as a model of call center, Applied Mathematical Modelling, 37 (2013), 958-976.  doi: 10.1016/j.apm.2012.03.021.  Google Scholar

[35]

C. Knessl and J. S. H. van Leeuwaarden, Transient analysis of the Erlang A model, Mathematical Methods of Operations Research, 82 (2015), 143-173.  doi: 10.1007/s00186-015-0498-9.  Google Scholar

[36]

R. Kumar and S. Sharma, Transient analysis of an M/M/c queuing system with balking and retention of reneging customers, Communications in Statistics - Theory and Methods, 47 (2018), 1318-1327.  doi: 10.1080/03610926.2017.1319485.  Google Scholar

[37]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytical Methods in Stochastic Modeling, ASA-SIAM Series on Statistics and Applied Probability. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; American Statistical Association, Alexandria, VA, 1999 doi: 10.1137/1.9780898719734.  Google Scholar

[38]

D. M. Lucantoni, New results for the single server queue with a batch Markovian arrival process, Comm. Statist. Stochastic Models, 7 (1991), 1-46.  doi: 10.1080/15326349108807174.  Google Scholar

[39]

D. M. LucantoniK. S. Meier-Hellstern and M. F. Neuts, A single server queue with server vacations and a class of nonrenewal arrival processes, Adv. Applied Prob., 22 (1990), 676-705.  doi: 10.2307/1427464.  Google Scholar

[40]

A. Mandelbaum and S. Zeltyn, The impact of customers' patience on delay and abandonment: Some empirically-driven experiments with the $M/M/ n + G$ queue, OR Spectrum, 26 (2004), 377-411.  doi: 10.1007/s00291-004-0164-8.  Google Scholar

[41]

M. MandjesD. Mitra and W. Scheinhardt, Models of network access using feedback fluid queues, Queueing Syst., 44 (2003), 365-398.  doi: 10.1023/A:1025147422141.  Google Scholar

[42]

K. Mitchell, Constructing a correlated sequence of matrix exponentials with invariant first-order properties, Operations Research Letters, 28 (2001), 27-34.  doi: 10.1016/S0167-6377(00)00062-6.  Google Scholar

[43]

A. Movaghar, On queueing with customer impatience until the beginning of service, Queueing Systems, 29 (1998), 337-350.  doi: 10.1023/A:1019196416987.  Google Scholar

[44]

M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, Dover Publications, Inc., 1981.  Google Scholar

[45]

M. F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications, Marcel Dekker, N.Y., 1989.  Google Scholar

[46]

H. OkamuraT. Dohi and K. S. Trivedi, Markovian arrival process parameter estimation with group data, IEEE/ACM Trans. Netw., 17 (2009), 1326-1339.  doi: 10.1109/TNET.2008.2008750.  Google Scholar

[47]

C. Palm, Methods of judging the annoyance caused by congestion, Tele, 4 (1953), 189-208.   Google Scholar

[48]

W. ScheinhardtN. Van Foreest and M. Mandjes, Continuous feedback fluid queues, Operations Research Letters, 33 (2005), 551-559.  doi: 10.1016/j.orl.2004.11.008.  Google Scholar

[49]

H. C. Tijms, Stochastic Modelling and Analysis: A Computational Approach, John Wiley & Sons, Ltd., Chichester, 1986.  Google Scholar

[50]

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Figure 1.  Venn diagram illustrating the relationship between CMFQs, MRMFQs, and CFFQs
Figure 2.  Sample path of the random process $ Z_f(t), t\geq 0 $
Figure 3.  Sample path of the random process $ \tilde{Z}_f(t), t\geq 0 $
Figure 4.  Sample path of the random process $Z_f(t), t\geq 0$
Table 1.  Results for the $ M/M/s+M $ scenario with $ s=10 $, $ \lambda=10 $, $ \mu=1 $, $ E(A)=1 $
Analysis $ K=10 $ Analysis $ K=50 $ Analysis $ K=250 $ Ref. [59] Simulation Results
$ \Pr \left\{ {W=0 } \right\} $ 0.46000 0.45802 0.45794 0.45793 $ 0.45829 \pm 0.00209 $
$ \Pr \left\{ {W=0|\mathcal{S} } \right\}$ 0.52612 0.52353 0.52343 0.52341 $ 0.52375 \pm 0.00195 $
$ \Pr \left\{ {\mathcal{A}} \right\} $ 0.12567 0.12513 0.12511 0.12511 $ 0.12498 \pm 0.00084 $
$ E(W|\mathcal{S}) $ 0.11648 0.11500 0.11494 0.11494 $ 0.11491 \pm 0.00077 $
$ \text{Var}(W|\mathcal{S}) $ 0.03377 0.03310 0.03307 0.03307 $ 0.03299 \pm 0.00030 $
$ F_{W|\mathcal{S},W >0}(.1) $ 0.28108 0.28100 0.28108 0.28107 $ 0.28060 \pm 0.00093 $
$ F_{W|\mathcal{S},W >0}(.2) $ 0.51155 0.51285 0.51283 0.51283 $ 0.51208 \pm 0.00145 $
Analysis $ K=10 $ Analysis $ K=50 $ Analysis $ K=250 $ Ref. [59] Simulation Results
$ \Pr \left\{ {W=0 } \right\} $ 0.46000 0.45802 0.45794 0.45793 $ 0.45829 \pm 0.00209 $
$ \Pr \left\{ {W=0|\mathcal{S} } \right\}$ 0.52612 0.52353 0.52343 0.52341 $ 0.52375 \pm 0.00195 $
$ \Pr \left\{ {\mathcal{A}} \right\} $ 0.12567 0.12513 0.12511 0.12511 $ 0.12498 \pm 0.00084 $
$ E(W|\mathcal{S}) $ 0.11648 0.11500 0.11494 0.11494 $ 0.11491 \pm 0.00077 $
$ \text{Var}(W|\mathcal{S}) $ 0.03377 0.03310 0.03307 0.03307 $ 0.03299 \pm 0.00030 $
$ F_{W|\mathcal{S},W >0}(.1) $ 0.28108 0.28100 0.28108 0.28107 $ 0.28060 \pm 0.00093 $
$ F_{W|\mathcal{S},W >0}(.2) $ 0.51155 0.51285 0.51283 0.51283 $ 0.51208 \pm 0.00145 $
Table 2.  Comparison of analytical results with simulations for the $ MAP/M/s+M $ scenario
Analysis $ K=10 $} Analysis $ K=50 $} Analysis $ K=250 $} Simulation Results
$ \Pr \left\{ { W=0 } \right\} $ 0.43620 0.43464 0.43458 $ 0.43473 \pm 0.00097 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.51361 0.51152 0.51143 $ 0.51158 \pm 0.00096 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.15073 0.15029 0.15027 $ 0.15023 \pm 0.00043 $
$ E(W|\mathcal{S}) $ 0.14046 0.13743 0.13737 $ 0.13718 \pm 0.00043 $
$ \text{Var}(W|\mathcal{S}) $ 0.12746 0.04456 0.04451 $ 0.04437 \pm 0.00018 $
$ F_{W|\mathcal{S},W >0}(.1) $ 0.23845 0.23847 0.23854 $ 0.23854 \pm 0.00066 $
$ F_{W|\mathcal{S},W >0}(.2) $ 0.44512 0.44644 0.44642 $ 0.44661 \pm 0.00094 $
Analysis $ K=10 $} Analysis $ K=50 $} Analysis $ K=250 $} Simulation Results
$ \Pr \left\{ { W=0 } \right\} $ 0.43620 0.43464 0.43458 $ 0.43473 \pm 0.00097 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.51361 0.51152 0.51143 $ 0.51158 \pm 0.00096 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.15073 0.15029 0.15027 $ 0.15023 \pm 0.00043 $
$ E(W|\mathcal{S}) $ 0.14046 0.13743 0.13737 $ 0.13718 \pm 0.00043 $
$ \text{Var}(W|\mathcal{S}) $ 0.12746 0.04456 0.04451 $ 0.04437 \pm 0.00018 $
$ F_{W|\mathcal{S},W >0}(.1) $ 0.23845 0.23847 0.23854 $ 0.23854 \pm 0.00066 $
$ F_{W|\mathcal{S},W >0}(.2) $ 0.44512 0.44644 0.44642 $ 0.44661 \pm 0.00094 $
Table 3.  Comparison of analytical results with simulations for the $ MAP/M/s+M_B $ scenario
Analysis $ K=10 $} Analysis $ K=50 $} Analysis $ K=250 $} Simulation Results
$ \Pr \left\{ { W=0 } \right\} $ 0.69025 0.68969 0.68968 $ 0.68977 \pm 0.00034 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.88516 0.88428 0.88425 $ 0.88430 \pm 0.00019 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.22020 0.22006 0.22006 $ 0.21999 \pm 0.00024 $
$ E(W|\mathcal{S}) $ 0.01563 0.01483 0.01480 $ 0.01481 \pm 0.00004 $
$ \text{Var}(W|\mathcal{S}) $ 0.00368 0.00344 0.00343 $ 0.00344 \pm 0.00001 $
$ F_{W|\mathcal{S},W>0}(.1) $ 0.53344 0.53371 0.53384 $ 0.53332\pm 0.00061 $
$ F_{W|\mathcal{S},W>0}(.2) $ 0.78709 0.78846 0.78846 $ 0.78832\pm 0.00055 $
Analysis $ K=10 $} Analysis $ K=50 $} Analysis $ K=250 $} Simulation Results
$ \Pr \left\{ { W=0 } \right\} $ 0.69025 0.68969 0.68968 $ 0.68977 \pm 0.00034 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.88516 0.88428 0.88425 $ 0.88430 \pm 0.00019 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.22020 0.22006 0.22006 $ 0.21999 \pm 0.00024 $
$ E(W|\mathcal{S}) $ 0.01563 0.01483 0.01480 $ 0.01481 \pm 0.00004 $
$ \text{Var}(W|\mathcal{S}) $ 0.00368 0.00344 0.00343 $ 0.00344 \pm 0.00001 $
$ F_{W|\mathcal{S},W>0}(.1) $ 0.53344 0.53371 0.53384 $ 0.53332\pm 0.00061 $
$ F_{W|\mathcal{S},W>0}(.2) $ 0.78709 0.78846 0.78846 $ 0.78832\pm 0.00055 $
Table 4.  Comparison of analytical results with simulations for the $ MAP/M/s+HE $ scenario
Analysis $ K=10 $} Analysis $ K=50 $} Analysis $ K=250 $} Simulation Results
$ \Pr \left\{ { W=0 } \right\} $ 0.39314 0.39104 0.39096 $ 0.39136 \pm 0.00088 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.45551 0.45276 0.45266 $ 0.45311 \pm 0.00087 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.13692 0.13632 0.13630 $ 0.13629 \pm 0.00037 $
$ E(W|\mathcal{S}) $ 0.20596 0.20195 0.20180 $ 0.20175 \pm 0.00061 $
$ \text{Var}(W|\mathcal{S}) $ 0.08667 0.08281 0.08266 $ 0.08266 \pm 0.00039 $
$ F_{W|\mathcal{S},W>0}(.1) $ 0.18329 0.18628 0.18639 $ 0.18610\pm 0.00054 $
$ F_{W|\mathcal{S},W>0}(.2) $ 0.35371 0.35537 0.35547 $ 0.35501\pm 0.00085 $
Analysis $ K=10 $} Analysis $ K=50 $} Analysis $ K=250 $} Simulation Results
$ \Pr \left\{ { W=0 } \right\} $ 0.39314 0.39104 0.39096 $ 0.39136 \pm 0.00088 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.45551 0.45276 0.45266 $ 0.45311 \pm 0.00087 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.13692 0.13632 0.13630 $ 0.13629 \pm 0.00037 $
$ E(W|\mathcal{S}) $ 0.20596 0.20195 0.20180 $ 0.20175 \pm 0.00061 $
$ \text{Var}(W|\mathcal{S}) $ 0.08667 0.08281 0.08266 $ 0.08266 \pm 0.00039 $
$ F_{W|\mathcal{S},W>0}(.1) $ 0.18329 0.18628 0.18639 $ 0.18610\pm 0.00054 $
$ F_{W|\mathcal{S},W>0}(.2) $ 0.35371 0.35537 0.35547 $ 0.35501\pm 0.00085 $
Table 5.  Comparison of analytical results with simulations for the $ MAP/M/s+D $ scenario
Analysis Simulation Results
$ \Pr \left\{ { W=0} \right\} $ 0.37989 $ 0.38040 \pm 0.00091 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.43851 $ 0.43893 \pm 0.00090 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.13367 $ 0.13334 \pm 0.00042 $
$ E(W|\mathcal{S}) $ 0.14990 $ 0.14974 \pm 0.00033 $
$ \text{Var}(W|\mathcal{S}) $ 0.02964 $ 0.02962 \pm 0.00004 $
$ F_{W|\mathcal{S},W>0}(.1) $ 0.17763 $ 0.17778 \pm 0.00047 $
$ F_{W|\mathcal{S},W>0}(.2) $ 0.35825 $ 0.35855 \pm 0.00072 $
Analysis Simulation Results
$ \Pr \left\{ { W=0} \right\} $ 0.37989 $ 0.38040 \pm 0.00091 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.43851 $ 0.43893 \pm 0.00090 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.13367 $ 0.13334 \pm 0.00042 $
$ E(W|\mathcal{S}) $ 0.14990 $ 0.14974 \pm 0.00033 $
$ \text{Var}(W|\mathcal{S}) $ 0.02964 $ 0.02962 \pm 0.00004 $
$ F_{W|\mathcal{S},W>0}(.1) $ 0.17763 $ 0.17778 \pm 0.00047 $
$ F_{W|\mathcal{S},W>0}(.2) $ 0.35825 $ 0.35855 \pm 0.00072 $
Table 6.  Comparison of analytical results with simulations for the $ MAP/M/s+W $ scenario
Analysis $ K=10 $ Analysis $ K=50 $ Analysis $ K=250 $ Simulation Results
$ \Pr \left\{ { W=0 } \right\} $ 0.34776 0.33324 0.33138 $ 0.33166 \pm 0.00097 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.39620 0.37774 0.37538 $ 0.37557 \pm 0.00098 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.12227 0.11778 0.11721 $ 0.11693 \pm 0.00042 $
$ E(W|\mathcal{S}) $ 0.31926 0.25150 0.25138 $ 0.25104 \pm 0.00065 $
$ \text{Var}(W|\mathcal{S}) $ 3.91308 0.12642 0.08071 $ 0.07974 \pm 0.00018 $
$ F_{W|\mathcal{S},W>0}(.1) $ 0.13862 0.13371 0.13361 $ 0.13401 \pm 0.00046 $
$ F_{W|\mathcal{S},W>0}(.2) $ 0.27149 0.26618 0.26670 $ 0.26723 \pm 0.00070 $
Analysis $ K=10 $ Analysis $ K=50 $ Analysis $ K=250 $ Simulation Results
$ \Pr \left\{ { W=0 } \right\} $ 0.34776 0.33324 0.33138 $ 0.33166 \pm 0.00097 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.39620 0.37774 0.37538 $ 0.37557 \pm 0.00098 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.12227 0.11778 0.11721 $ 0.11693 \pm 0.00042 $
$ E(W|\mathcal{S}) $ 0.31926 0.25150 0.25138 $ 0.25104 \pm 0.00065 $
$ \text{Var}(W|\mathcal{S}) $ 3.91308 0.12642 0.08071 $ 0.07974 \pm 0.00018 $
$ F_{W|\mathcal{S},W>0}(.1) $ 0.13862 0.13371 0.13361 $ 0.13401 \pm 0.00046 $
$ F_{W|\mathcal{S},W>0}(.2) $ 0.27149 0.26618 0.26670 $ 0.26723 \pm 0.00070 $
Table 7.  Comparison of analytical results with simulations for the $ MAP/M/s+E_2 $ scenario
Analysis $ K=10 $ Analysis $ K=50 $ Analysis $ K=250 $ Simulation Results
$ \Pr \left\{ { W=0 } \right\} $ 0.30664 0.29490 0.29380 $ 0.29336 \pm 0.00115 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.34379 0.32928 0.32793 $ 0.32737 \pm 0.00115 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.10806 0.10442 0.10408 $ 0.10393 \pm 0.00048 $
$ E(W|\mathcal{S}) $ 0.37254 0.37450 0.37482 $ 0.37475 \pm 0.00138 $
$ \text{Var}(W|\mathcal{S}) $ 0.17112 0.17499 0.17560 $ 0.17527 \pm 0.00075 $
$ F_{W|\mathcal{S},W>0}(.1) $ 0.109761 0.10744 0.10785 $ 0.10782 \pm 0.00047 $
$ F_{W|\mathcal{S},W>0}(.2) $ 0.21430 0.21334 0.21375 $ 0.21371 \pm 0.00082 $
Analysis $ K=10 $ Analysis $ K=50 $ Analysis $ K=250 $ Simulation Results
$ \Pr \left\{ { W=0 } \right\} $ 0.30664 0.29490 0.29380 $ 0.29336 \pm 0.00115 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.34379 0.32928 0.32793 $ 0.32737 \pm 0.00115 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.10806 0.10442 0.10408 $ 0.10393 \pm 0.00048 $
$ E(W|\mathcal{S}) $ 0.37254 0.37450 0.37482 $ 0.37475 \pm 0.00138 $
$ \text{Var}(W|\mathcal{S}) $ 0.17112 0.17499 0.17560 $ 0.17527 \pm 0.00075 $
$ F_{W|\mathcal{S},W>0}(.1) $ 0.109761 0.10744 0.10785 $ 0.10782 \pm 0.00047 $
$ F_{W|\mathcal{S},W>0}(.2) $ 0.21430 0.21334 0.21375 $ 0.21371 \pm 0.00082 $
Table 8.  Comparison of analytical results with simulations for the $ MAP/M/s+E_3 $ scenario
Analysis $ K=10 $ Analysis $ K=50 $ Analysis $ K=250 $ Simulation Results
$ \Pr \left\{ { W=0 } \right\} $ 0.24365 0.22187 0.21925 $ 0.21946 \pm 0.00106 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.26659 0.24087 0.23780 $ 0.23791 \pm 0.00108 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.08603 0.07886 0.07801 $ 0.07757 \pm 0.00044 $
$ E(W|\mathcal{S}) $ 0.63951 0.65123 0.65284 $ 0.65137 \pm 0.00222 $
$ \text{Var}(W|\mathcal{S}) $ 0.34263 0.38303 0.38981 $ 0.39046 \pm 0.00157 $
$ F_{W|\mathcal{S},W>0}(.1) $ 0.07563 0.06845 0.06783 $ 0.06795 \pm 0.00035 $
$ F_{W|\mathcal{S},W>0}(.2) $ 0.14729 0.13513 0.13433 $ 0.13458 \pm 0.00061 $
Analysis $ K=10 $ Analysis $ K=50 $ Analysis $ K=250 $ Simulation Results
$ \Pr \left\{ { W=0 } \right\} $ 0.24365 0.22187 0.21925 $ 0.21946 \pm 0.00106 $
$ \Pr \left\{ { W=0|\mathcal{S}} \right\} $ 0.26659 0.24087 0.23780 $ 0.23791 \pm 0.00108 $
$ \Pr \left\{ { \mathcal{A} } \right\} $ 0.08603 0.07886 0.07801 $ 0.07757 \pm 0.00044 $
$ E(W|\mathcal{S}) $ 0.63951 0.65123 0.65284 $ 0.65137 \pm 0.00222 $
$ \text{Var}(W|\mathcal{S}) $ 0.34263 0.38303 0.38981 $ 0.39046 \pm 0.00157 $
$ F_{W|\mathcal{S},W>0}(.1) $ 0.07563 0.06845 0.06783 $ 0.06795 \pm 0.00035 $
$ F_{W|\mathcal{S},W>0}(.2) $ 0.14729 0.13513 0.13433 $ 0.13458 \pm 0.00061 $
Table 9.  The cdf of the virtual waiting time evaluated at time $ \tau $, $ F_v^{a,b,\pi_0,\theta_0}(\tau) $, obtained by the proposed method using Erlang-$ \ell $ and CME-$ \ell $ distributions for three values of the parameter $ \ell $, and for the scenario $ \theta_0 = [\begin{smallmatrix} 1 & 0 \end{smallmatrix}] $
Erlang-$ \ell $ CME-$ \ell $
$ \tau $ $ b $ Simulation $ \ell=25 $ $ \ell=51 $ $ \ell=101 $ $ \ell=25 $ $ \ell=51 $ $ \ell=101 $
$ 1/4 $ 0.61906$ \pm $0.00027 0.59251 0.60529 0.61200 0.61705 0.61883 0.61922
$ 1/2 $ 0.39325$ \pm $0.00050 0.38744 0.39032 0.39185 0.39309 0.39345 0.39354
$ 1 $ $ 1 $ 0.11791$ \pm $0.00065 0.13245 0.12551 0.12189 0.11922 0.11822 0.11800
$ 2 $ 0.00323$ \pm $0.00044 0.00607 0.00456 0.00388 0.00345 0.00328 0.00324
$ 4 $ 0.00000$ \pm $0.00002 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
$ 1/4 $ 0.97700$ \pm $0.00022 0.97692 0.97698 0.97700 0.97690 0.97699 0.97700
$ 1/2 $ 0.96964$ \pm $0.00029 0.96910 0.96949 0.96960 0.96953 0.96965 0.96967
$ 5 $ $ 1 $ 0.95339$ \pm $0.00031 0.94898 0.95163 0.95257 0.95294 0.95326 0.95331
$ 2 $ 0.86128$ \pm $0.00039 0.82707 0.84443 0.85283 0.85859 0.86077 0.86124
$ 4 $ 0.14516$ \pm $0.00050 0.16177 0.15408 0.14986 0.14667 0.14544 0.14517
$ 1/4 $ 0.98695$ \pm $0.00016 0.98684 0.98691 0.98694 0.98693 0.84307 0.98697
$ 1/2 $ 0.98274$ \pm $0.00020 0.98261 0.98270 0.98274 0.98273 0.98278 0.98278
$ 25 $ $ 1 $ 0.97408$ \pm $0.00022 0.97381 0.97394 0.97400 0.97399 0.97405 0.97406
$ 2 $ 0.95078$ \pm $0.00035 0.95051 0.95073 0.95084 0.95084 0.95093 0.95094
$ 4 $ 0.86036$ \pm $0.00035 0.85903 0.85979 0.86006 0.86011 0.86027 0.86030
Erlang-$ \ell $ CME-$ \ell $
$ \tau $ $ b $ Simulation $ \ell=25 $ $ \ell=51 $ $ \ell=101 $ $ \ell=25 $ $ \ell=51 $ $ \ell=101 $
$ 1/4 $ 0.61906$ \pm $0.00027 0.59251 0.60529 0.61200 0.61705 0.61883 0.61922
$ 1/2 $ 0.39325$ \pm $0.00050 0.38744 0.39032 0.39185 0.39309 0.39345 0.39354
$ 1 $ $ 1 $ 0.11791$ \pm $0.00065 0.13245 0.12551 0.12189 0.11922 0.11822 0.11800
$ 2 $ 0.00323$ \pm $0.00044 0.00607 0.00456 0.00388 0.00345 0.00328 0.00324
$ 4 $ 0.00000$ \pm $0.00002 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
$ 1/4 $ 0.97700$ \pm $0.00022 0.97692 0.97698 0.97700 0.97690 0.97699 0.97700
$ 1/2 $ 0.96964$ \pm $0.00029 0.96910 0.96949 0.96960 0.96953 0.96965 0.96967
$ 5 $ $ 1 $ 0.95339$ \pm $0.00031 0.94898 0.95163 0.95257 0.95294 0.95326 0.95331
$ 2 $ 0.86128$ \pm $0.00039 0.82707 0.84443 0.85283 0.85859 0.86077 0.86124
$ 4 $ 0.14516$ \pm $0.00050 0.16177 0.15408 0.14986 0.14667 0.14544 0.14517
$ 1/4 $ 0.98695$ \pm $0.00016 0.98684 0.98691 0.98694 0.98693 0.84307 0.98697
$ 1/2 $ 0.98274$ \pm $0.00020 0.98261 0.98270 0.98274 0.98273 0.98278 0.98278
$ 25 $ $ 1 $ 0.97408$ \pm $0.00022 0.97381 0.97394 0.97400 0.97399 0.97405 0.97406
$ 2 $ 0.95078$ \pm $0.00035 0.95051 0.95073 0.95084 0.95084 0.95093 0.95094
$ 4 $ 0.86036$ \pm $0.00035 0.85903 0.85979 0.86006 0.86011 0.86027 0.86030
Table 10.  The cdf of the virtual waiting time evaluated at time $ \tau $, $ F_v^{a,b,\pi_0,\theta_0}(\tau) $, obtained by the proposed method using Erlang-$ \ell $ and CME-$ \ell $ distributions for three values of the parameter $ \ell $, and for the scenario $ \theta_0 = [\begin{smallmatrix} 0 & 1 \end{smallmatrix}] $
Erlang-$ \ell $ CME-$ \ell $
$ \tau $ $ b $ Simulation $ \ell=25 $ $ \ell=51 $ $ \ell=101 $ $ \ell=25 $ $ \ell=51 $ $ \ell=101 $
$ 1/4 $ 0.00663$ \pm $0.00010 0.00703 0.00681 0.00671 0.00664 0.00661 0.00660
$ 1/2 $ 0.00355$ \pm $0.00006 0.00400 0.00377 0.00365 0.00357 0.00354 0.00353
$ 1 $ $ 1 $ 0.00081$ \pm $0.00003 0.00111 0.00096 0.00089 0.00084 0.00082 0.00082
$ 2 $ 0.00002$ \pm $0.00001 0.00004 0.00002 0.00002 0.00002 0.00002 0.00002
$ 4 $ 0.00000$ \pm $0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
$ 1/4 $ 0.11115$ \pm $0.00032 0.11067 0.11085 0.11094 0.11100 0.11102 0.11103
$ 1/2 $ 0.10346$ \pm $0.00031 0.10305 0.10323 0.10332 0.10338 0.10340 0.10341
$ 5 $ $ 1 $ 0.08827$ \pm $0.00026 0.08794 0.08809 0.08816 0.08823 0.08824 0.08825
$ 2 $ 0.05324$ \pm $0.00020 0.05382 0.05349 0.05334 0.05326 0.05322 0.05321
$ 4 $ 0.00365$ \pm $0.00005 0.00538 0.00453 0.00412 0.00382 0.00372 0.00369
$ 1/4 $ 0.49669$ \pm $0.00059 0.49161 0.49415 0.49537 0.49620 0.49653 0.49660
$ 1/2 $ 0.48996$ \pm $0.00058 0.48497 0.48751 0.48872 0.48956 0.48989 0.48996
$ 25 $ $ 1 $ 0.47661$ \pm $0.00058 0.47159 0.47410 0.47531 0.47614 0.47647 0.47654
$ 2 $ 0.44349$ \pm $0.00059 0.43858 0.44104 0.44223 0.44304 0.44336 0.44343
$ 4 $ 0.33702$ \pm $0.00049 0.33222 0.33441 0.33547 0.33622 0.33649 0.33654
Erlang-$ \ell $ CME-$ \ell $
$ \tau $ $ b $ Simulation $ \ell=25 $ $ \ell=51 $ $ \ell=101 $ $ \ell=25 $ $ \ell=51 $ $ \ell=101 $
$ 1/4 $ 0.00663$ \pm $0.00010 0.00703 0.00681 0.00671 0.00664 0.00661 0.00660
$ 1/2 $ 0.00355$ \pm $0.00006 0.00400 0.00377 0.00365 0.00357 0.00354 0.00353
$ 1 $ $ 1 $ 0.00081$ \pm $0.00003 0.00111 0.00096 0.00089 0.00084 0.00082 0.00082
$ 2 $ 0.00002$ \pm $0.00001 0.00004 0.00002 0.00002 0.00002 0.00002 0.00002
$ 4 $ 0.00000$ \pm $0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
$ 1/4 $ 0.11115$ \pm $0.00032 0.11067 0.11085 0.11094 0.11100 0.11102 0.11103
$ 1/2 $ 0.10346$ \pm $0.00031 0.10305 0.10323 0.10332 0.10338 0.10340 0.10341
$ 5 $ $ 1 $ 0.08827$ \pm $0.00026 0.08794 0.08809 0.08816 0.08823 0.08824 0.08825
$ 2 $ 0.05324$ \pm $0.00020 0.05382 0.05349 0.05334 0.05326 0.05322 0.05321
$ 4 $ 0.00365$ \pm $0.00005 0.00538 0.00453 0.00412 0.00382 0.00372 0.00369
$ 1/4 $ 0.49669$ \pm $0.00059 0.49161 0.49415 0.49537 0.49620 0.49653 0.49660
$ 1/2 $ 0.48996$ \pm $0.00058 0.48497 0.48751 0.48872 0.48956 0.48989 0.48996
$ 25 $ $ 1 $ 0.47661$ \pm $0.00058 0.47159 0.47410 0.47531 0.47614 0.47647 0.47654
$ 2 $ 0.44349$ \pm $0.00059 0.43858 0.44104 0.44223 0.44304 0.44336 0.44343
$ 4 $ 0.33702$ \pm $0.00049 0.33222 0.33441 0.33547 0.33622 0.33649 0.33654
Table 11.  The cdf of the actual waiting time evaluated at time $ \tau $, $ F_a^{a,b,\pi_0,\theta_0}(\tau) $, obtained by the proposed method using Erlang-$ \ell $ and CME-$ \ell $ distributions for three values of the parameter $ \ell $, and for the scenario $ \theta_0 = [\begin{smallmatrix} 1 & 0 \end{smallmatrix}] $
Erlang-$ \ell $ CME-$ \ell $
$ \tau $ $ b $ Simulation $ \ell=25 $ $ \ell=51 $ $ \ell=101 $ $ \ell=25 $ $ \ell=51 $ $ \ell=101 $
$ 1/4 $ 0.52092$ \pm $0.00051 0.50227 0.51121 0.51598 0.51964 0.52089 0.52117
$ 1/2 $ 0.31345$ \pm $0.00049 0.31514 0.31438 0.31397 0.31375 0.31358 0.31355
$ 1 $ $ 1 $ 0.08271$ \pm $0.00028 0.09792 0.09062 0.08685 0.08408 0.08305 0.08282
$ 2 $ 0.00184$ \pm $0.00004 0.00382 0.00273 0.00226 0.00197 0.00186 0.00183
$ 4 $ 0.00000$ \pm $0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
$ 1/4 $ 0.97418$ \pm $0.00020 0.97393 0.97406 0.97409 0.97400 0.97409 0.97411
$ 1/2 $ 0.96674$ \pm $0.00023 0.96574 0.96635 0.96652 0.96649 0.96662 0.96664
$ 5 $ $ 1 $ 0.94941$ \pm $0.00023 0.94345 0.94692 0.94819 0.94876 0.94915 0.94922
$ 2 $ 0.84264$ \pm $0.00052 0.80496 0.82380 0.83310 0.83960 0.84204 0.84256
$ 4 $ 0.11830$ \pm $0.00040 0.13624 0.12782 0.12323 0.11975 0.11844 0.11815
$ 1/4 $ 0.98533$ \pm $0.00015 0.98517 0.98524 0.98528 0.98527 0.98530 0.98531
$ 1/2 $ 0.98108$ \pm $0.00018 0.98089 0.98098 0.98103 0.98102 0.98106 0.98107
$ 25 $ $ 1 $ 0.97205$ \pm $0.00017 0.97180 0.97193 0.97200 0.97199 0.97205 0.97206
$ 2 $ 0.94819$ \pm $0.00020 0.94767 0.94789 0.94800 0.94801 0.94810 0.94811
$ 4 $ 0.85268$ \pm $0.00038 0.85115 0.85203 0.85234 0.85240 0.85257 0.85260
Erlang-$ \ell $ CME-$ \ell $
$ \tau $ $ b $ Simulation $ \ell=25 $ $ \ell=51 $ $ \ell=101 $ $ \ell=25 $ $ \ell=51 $ $ \ell=101 $
$ 1/4 $ 0.52092$ \pm $0.00051 0.50227 0.51121 0.51598 0.51964 0.52089 0.52117
$ 1/2 $ 0.31345$ \pm $0.00049 0.31514 0.31438 0.31397 0.31375 0.31358 0.31355
$ 1 $ $ 1 $ 0.08271$ \pm $0.00028 0.09792 0.09062 0.08685 0.08408 0.08305 0.08282
$ 2 $ 0.00184$ \pm $0.00004 0.00382 0.00273 0.00226 0.00197 0.00186 0.00183
$ 4 $ 0.00000$ \pm $0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
$ 1/4 $ 0.97418$ \pm $0.00020 0.97393 0.97406 0.97409 0.97400 0.97409 0.97411
$ 1/2 $ 0.96674$ \pm $0.00023 0.96574 0.96635 0.96652 0.96649 0.96662 0.96664
$ 5 $ $ 1 $ 0.94941$ \pm $0.00023 0.94345 0.94692 0.94819 0.94876 0.94915 0.94922
$ 2 $ 0.84264$ \pm $0.00052 0.80496 0.82380 0.83310 0.83960 0.84204 0.84256
$ 4 $ 0.11830$ \pm $0.00040 0.13624 0.12782 0.12323 0.11975 0.11844 0.11815
$ 1/4 $ 0.98533$ \pm $0.00015 0.98517 0.98524 0.98528 0.98527 0.98530 0.98531
$ 1/2 $ 0.98108$ \pm $0.00018 0.98089 0.98098 0.98103 0.98102 0.98106 0.98107
$ 25 $ $ 1 $ 0.97205$ \pm $0.00017 0.97180 0.97193 0.97200 0.97199 0.97205 0.97206
$ 2 $ 0.94819$ \pm $0.00020 0.94767 0.94789 0.94800 0.94801 0.94810 0.94811
$ 4 $ 0.85268$ \pm $0.00038 0.85115 0.85203 0.85234 0.85240 0.85257 0.85260
Table 12.  The cdf of the actual waiting time evaluated at time $ \tau $, $ F_a^{a,b,\pi_0,\theta_0}(\tau) $, obtained by the proposed method using Erlang-$ \ell $ and CME-$ \ell $ distributions for three values of the parameter $ \ell $, and for the scenario $ \theta_0 = [\begin{smallmatrix} 0 & 1 \end{smallmatrix}] $
Erlang-$ \ell $ CME-$ \ell $
$ \tau $ $ b $ Simulation $ \ell=25 $ $ \ell=51 $ $ \ell=101 $ $ \ell=25 $ $ \ell=51 $ $ \ell=101 $
$ 1/4 $ 0.00502$ \pm $0.00005 0.00553 0.00529 0.00517 0.00509 0.00506 0.00505
$ 1/2 $ 0.00257$ \pm $0.00004 0.00307 0.00283 0.00271 0.00263 0.00260 0.00260
$ 1 $ $ 1 $ 0.00053$ \pm $0.00003 0.00078 0.00066 0.00060 0.00056 0.00054 0.00054
$ 2 $ 0.00001$ \pm $0.00000 0.00002 0.00002 0.00001 0.00001 0.00001 0.00001
$ 4 $ 0.00000$ \pm $0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
$ 1/4 $ 0.10802$ \pm $0.00030 0.10761 0.10780 0.10789 0.10795 0.10797 0.10798
$ 1/2 $ 0.10030$ \pm $0.00029 0.10000 0.10018 0.10026 0.10033 0.10035 0.10035
$ 5 $ $ 1 $ 0.08481$ \pm $0.00026 0.08463 0.08476 0.08483 0.08489 0.08491 0.08491
$ 2 $ 0.04944$ \pm $0.00022 0.05027 0.04985 0.04966 0.04954 0.04949 0.04948
$ 4 $ 0.00283$ \pm $0.00007 0.00438 0.00362 0.00324 0.00298 0.00289 0.00287
$ 1/4 $ 0.49362$ \pm $0.00051 0.48896 0.49149 0.49271 0.49355 0.49387 0.49394
$ 1/2 $ 0.48698$ \pm $0.00051 0.48230 0.48483 0.48605 0.48688 0.48721 0.48727
$ 25 $ $ 1 $ 0.47332$ \pm $0.00051 0.46860 0.47111 0.47232 0.47315 0.47347 0.47354
$ 2 $ 0.43941$ \pm $0.00051 0.43475 0.43720 0.43839 0.43920 0.43951 0.43958
$ 4 $ 0.32884$ \pm $0.00048 0.32438 0.32654 0.32759 0.32832 0.32859 0.32865
Erlang-$ \ell $ CME-$ \ell $
$ \tau $ $ b $ Simulation $ \ell=25 $ $ \ell=51 $ $ \ell=101 $ $ \ell=25 $ $ \ell=51 $ $ \ell=101 $
$ 1/4 $ 0.00502$ \pm $0.00005 0.00553 0.00529 0.00517 0.00509 0.00506 0.00505
$ 1/2 $ 0.00257$ \pm $0.00004 0.00307 0.00283 0.00271 0.00263 0.00260 0.00260
$ 1 $ $ 1 $ 0.00053$ \pm $0.00003 0.00078 0.00066 0.00060 0.00056 0.00054 0.00054
$ 2 $ 0.00001$ \pm $0.00000 0.00002 0.00002 0.00001 0.00001 0.00001 0.00001
$ 4 $ 0.00000$ \pm $0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
$ 1/4 $ 0.10802$ \pm $0.00030 0.10761 0.10780 0.10789 0.10795 0.10797 0.10798
$ 1/2 $ 0.10030$ \pm $0.00029 0.10000 0.10018 0.10026 0.10033 0.10035 0.10035
$ 5 $ $ 1 $ 0.08481$ \pm $0.00026 0.08463 0.08476 0.08483 0.08489 0.08491 0.08491
$ 2 $ 0.04944$ \pm $0.00022 0.05027 0.04985 0.04966 0.04954 0.04949 0.04948
$ 4 $ 0.00283$ \pm $0.00007 0.00438 0.00362 0.00324 0.00298 0.00289 0.00287
$ 1/4 $ 0.49362$ \pm $0.00051 0.48896 0.49149 0.49271 0.49355 0.49387 0.49394
$ 1/2 $ 0.48698$ \pm $0.00051 0.48230 0.48483 0.48605 0.48688 0.48721 0.48727
$ 25 $ $ 1 $ 0.47332$ \pm $0.00051 0.46860 0.47111 0.47232 0.47315 0.47347 0.47354
$ 2 $ 0.43941$ \pm $0.00051 0.43475 0.43720 0.43839 0.43920 0.43951 0.43958
$ 4 $ 0.32884$ \pm $0.00048 0.32438 0.32654 0.32759 0.32832 0.32859 0.32865
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