# Sensitivity Analysis and Simulation of a Multiserver Queueing System with Mixed Service Time Distribution

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## Abstract

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## 1. Introduction

## 2. Two-Component Mixture Distributions

## 3. Multiserver System Sensitivity

**Lemma**

**1.**

## 4. Exact Steady-State Simulation by Regenerative Approach

- sample stochastic copies ${V}^{\left(k\right)}=({V}_{1}^{\left(k\right)},{V}_{2}^{\left(k\right)},\dots ),\phantom{\rule{0.166667em}{0ex}}k=1,2,\dots $ of the sequence of workload vectors using recursion Equation (28); each sequence starts with ${V}_{1}^{\left(k\right)}=0$ and lasts until the event ${V}_{\tau \left(k\right)}^{\left(k\right)}=0$ happens at some instant $\tau \left(k\right)$; note that $\left\{\tau \right(k\left)\right\}$ are iid random variables distributed as a generic regeneration period $\tau $ of RA system;
- repeat previous step until the event $\tau \left(j\right)>{\tau}_{e}$ happens in some sample ${V}^{\left(j\right)}=({V}_{1}^{\left(j\right)},{V}_{2}^{\left(j\right)},\dots )$; and,
- the value ${V}_{{\tau}_{e}}^{\left(j\right)}$ of the workload vector ${V}^{\left(j\right)}$ at instant ${\tau}_{e}$, has the target steady-state distribution of the workload in the original $M/G/c$ system.

## 5. Simulation Results

#### 5.1. Experiment 1: Hyperexponential Case

#### 5.2. Experiment 2a: Pareto Case, Sensitivity to Mixing Parameter

#### 5.3. Experiment 2b: Pareto Case, Sensitivity to Contaminating Distribution

#### 5.4. Experiment 2c: Pareto Case, Constant Load

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

d.f. | distribution function |

w.p. | with probability |

## References

- Al-Hussaini, E.K.; Sultan, K.S. Reliability and hazard based on finite mixture models. In Handbook of Statistics; Elsevier: Amsterdam, The Netherlands, 2001; Volume 20, pp. 139–183. [Google Scholar] [CrossRef]
- Shaked, M.; Spizzichino, F. Mixtures and monotonicity of failure rate functions. In Handbook of Statistics; Elsevier: Amsterdam, The Netherlands, 2001; Volume 20, pp. 185–198. [Google Scholar] [CrossRef]
- Sevast’yanov, B.A. An Ergodic Theorem for Markov Processes and Its Application to Telephone Systems with Refusals. Theory Probab. Appl.
**1957**, 2, 104–112. [Google Scholar] [CrossRef] - Kalashnikov, V.V. Stability Analysis of in Queueing Problems by a Method of Trial Functions. Theory Probab. Appl.
**1977**, 22, 86–103. [Google Scholar] [CrossRef] - Müller, A.; Stoyan, D. Comparison Methods for Stochastic Models and Risks; Wiley Series in Probability and Statistics; Wiley: Hoboken, NJ, USA, 2002. [Google Scholar]
- Zolotarev, V.M. On the stochastic continuity of the queuing systems of type G|G|1. Theory Probab. Appl.
**1977**, 21, 250–269. [Google Scholar] [CrossRef] - Zolotarev, V.M. Quantitative estimates for the continuity property of queueing systems of type G|G|∞. Theory Probab. Appl.
**1978**, 22, 679–691. [Google Scholar] [CrossRef] - Zolotarev, V.M. Qualitative Estimates in Problems of Continuity of Queuing Systems. Theory Probab. Appl.
**1975**, 20, 211–213. [Google Scholar] [CrossRef] - Batrakova, D.; Korolev, V.; Shorgin, S. A new method for the probabilistic and statistical analysis of information flows in telecommunication networks. Inform. Appl.
**2007**, 1, 40–53. [Google Scholar] - Daley, D.J. Queueing Output Processes. Adv. Appl. Probab.
**1976**, 8, 395. [Google Scholar] [CrossRef] - Daley, D.J. Revisiting queueing output processes: A point process viewpoint. Queueing Syst.
**2011**, 68, 395–405. [Google Scholar] [CrossRef] - Korolev, V.Y.E.; Krylov, V.A.; Kuz’min, V.Y.E. Stability of finite mixtures of generalized Gamma-distributions with respect to disturbance of parameters. Inform. Appl.
**2011**, 5, 31–38. [Google Scholar] - Kalashnikov, V.V.; Tsitsiashvili, G.S. Stability analysis of queueing systems. J. Sov. Math.
**1981**, 17, 2238–2255. [Google Scholar] [CrossRef] - McLachlan, G.J.; Lee, S.X.; Rathnayake, S.I. Finite Mixture Models. Annu. Rev. Stat. Its Appl.
**2019**, 6, 355–378. [Google Scholar] [CrossRef] - Sigman, K. Exact simulation of the stationary distribution of the FIFO M/G/c queue: The general case for ρ < c. Queueing Syst.
**2012**, 70, 37–43. [Google Scholar] [CrossRef] - Feitelson, D.G. Workload Modeling for Computer Dystems Performance Evaluation; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Morozov, E.; Peshkova, I.; Rumyantsev, A. On Failure Rate Comparison of Finite Multiserver Systems. In Distributed Computer and Communication Networks; Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V., Eds.; Springer International Publishing: Cham, Germany, 2019; Volume 11965, pp. 419–431. [Google Scholar] [CrossRef]
- Marshall, A.W.; Olkin, I. Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families; Springer Series in Statistics; Springer: New York, NY, USA; London, UK, 2007. [Google Scholar]
- Goldstein, M. Contamination Distributions. In The Annals of Statistics; Institute of Mathematical Statistics: Beachwood, OH, USA, 1982; Volume 10, pp. 174–183. [Google Scholar]
- Block, H.W. The Failure Rates of Mixtures. In Advances in Distribution Theory, Order Statistics, and Inference; Balakrishnan, N., Sarabia, J.M., Castillo, E., Eds.; Birkhäuser Boston: Boston, MA, USA, 2006; pp. 267–277. [Google Scholar] [CrossRef]
- Barlow, R.E.; Proschan, F. Mathematical Theory of Reliability; Classics in Applied Mathematics; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1996. [Google Scholar] [CrossRef]
- Goldie, C.M.; Klüppelberg, C. Subexponential Distributions. In A Practical Guide to Heavy Tails: Statistical Techniques and Applications; Birkhauser Boston Inc.: Cambridge, MA, USA, 1998; pp. 435–459. [Google Scholar]
- Shaked, M. Bounds on the Distance of a Mixture from Its Parent Distribution. J. Appl. Probab.
**1981**, 18, 853–863. [Google Scholar] [CrossRef] - Asmussen, S. Applied Probability and Queues; Springer: New York, NY, USA, 2003. [Google Scholar]
- Kiefer, J.D.; Wolfowitz, J. On the theory of queues with many servers. Trans. Am. Math. Soc.
**1955**, 78, 1–18. [Google Scholar] [CrossRef] - Kleinrock, L. Theory, Volume 1, Queueing Systems; Wiley-Interscience: Hoboken, NJ, USA, 1975. [Google Scholar]
- Whitt, W. Comparing counting processes and queues. Adv. Appl. Probab.
**1981**, 13, 207–220. [Google Scholar] [CrossRef] - Thorrison, H. Coupling, Stationarity, and Regeneration; Springer: New York, NY, USA, 2000. [Google Scholar]
- Shaked, M.; Shanthikumar, J.G. Stochastic Orders; Springer Series in Statistics; Springer: New York, NY, USA, 2007. [Google Scholar]
- Whitt, W. Approximations for the GI/G/M Queue. Prod. Oper. Manag.
**1993**, 2, 114–161. [Google Scholar] [CrossRef] - Van Hoorn, M.; Tijms, H. Approximations for the waiting time distribution of the M/G/c queue. Perform. Eval.
**1982**, 2, 22–28. [Google Scholar] [CrossRef] - Ma, B.N.W.; Mark, J.W. Approximation of the Mean Queue Length of an M/G/c Queueing System. Oper. Res.
**1995**, 43, 158–165. [Google Scholar] [CrossRef] - Kimura, T. Approximations for multi-server queues: System interpolations. Queueing Syst.
**1994**, 17, 347–382. [Google Scholar] [CrossRef] - Gupta, V.; Harchol-Balter, M.; Dai, J.G.; Zwart, B. On the inapproximability of M/G/K: Why two moments of job size distribution are not enough. Queueing Syst.
**2010**, 64, 5–48. [Google Scholar] [CrossRef] [Green Version] - Blanchet, J.; Pei, Y.; Sigman, K. Exact sampling for some multi-dimensional queueing models with renewal input. Adv. Appl. Probab.
**2019**, 51, 1179–1208. [Google Scholar] [CrossRef] [Green Version] - Xiong, Y. Perfect and Nearly Perfect Sampling of Work-Conserving Queues. Ph.D. Thesis, The School of Graduate and Postdoctoral Studies, The University of Western Ontario, London, ON, Canada, 2015. [Google Scholar]
- Blanchet, J.; Dong, J.; Pei, Y. Perfect Sampling of GI/GI/c Queues. arXiv
**2015**, arXiv:1508.02262. [Google Scholar] [CrossRef] [Green Version] - Nair, N.U.; Preeth, M. On some properties of equilibrium distributions of order n. Stat. Methods Appl.
**2009**, 18, 453–464. [Google Scholar] [CrossRef] - de Smit, J.H. A numerical solution for the multi-server queue with hyper-exponential service times. Oper. Res. Lett.
**1983**, 2, 217–224. [Google Scholar] [CrossRef] [Green Version] - Morozov, E.; Peshkova, I.; Rumyantsev, A. On Regenerative Envelopes for Cluster Model Simulation. In Proceedings of the Distributed Computer and Communication Networks: 19th International Conference, DCCN 2016, Moscow, Russia, 21–25 November 2016; Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V., Eds.; Springer International Publishing: Cham, Germany, 2016; pp. 222–230. [Google Scholar] [CrossRef]
- Rumyantsev, A.; Peshkova, I. Artificial Regeneration Based Regenerative Estimation of Multiserver System with Multiple Vacations Policy. In Information Technologies and Mathematical Modelling. Queueing Theory and Applications; Dudin, A., Nazarov, A., Moiseev, A., Eds.; Springer International Publishing: Berlin/Heidelberg, Germany, 2019; Volume 1109, pp. 38–50. [Google Scholar] [CrossRef]

**Figure 1.**The distance $\mathsf{\Delta}({F}_{M},{F}_{1})$ with mixing parameter $p=0.9$ and an upper bound $\delta ({\alpha}_{1},{\alpha}_{2})$ vs. parameter ${\alpha}_{2}$.

**Figure 2.**Theoretical distribution of the steady-state queue size in an $M/M/10$ system vs.empirical distribution ($N=5000$ samples), with input rate $\lambda =7.5$, service rate $\mu =1.5$. The uniform distance between the theoretical and estimated queue size distributions equals $\mathsf{\Delta}$ = 0.0091.

**Figure 3.**Distance, $\mathsf{\Delta}({\widehat{F}}_{{Q}^{\left(M\right)}},{\widehat{F}}_{{Q}^{\left(1\right)}})$, between the empirical queue size d.f. in a basic $M/M/5$ system with input rate $\lambda =5$, service rate ${\mu}_{1}=2$, compared to a contaminated $M/{H}_{2}/5$ system with input rate $\lambda =5$ and hyperexponential service times being a mixture with ${\mu}_{1}=2$ and ${\mu}_{2}=2,2.4,\dots ,8$, $p=0.7$, obtained from N = 10,000 samples, vs. service time d.f. distance, $\mathsf{\Delta}({F}_{M},{F}_{1})$.

**Figure 4.**Distance between the empirical queue size d.f. in a basic $M/G/c$ system with $c=4$, $\rho =0.5$, ${F}_{1}$ being $Pareto(2.1,1)$ service time d.f. and $\lambda =2.2$, and system with a mixture, ${F}_{M}$ of $Pareto(2.1,1)$ and $Pareto(4.9,1)$ service time d.f. vs. the distance between ${F}_{1}$ and ${F}_{M}$, for varying $p=1,0.95,\dots ,0.25$.

**Figure 5.**Distance between the empirical queue size d.f. in a basic $M/G/c$ system with $c=4$, $\rho =0.5$, ${F}_{1}$ being $Pareto(2.1,1)$ service time d.f. and $\lambda =2.2$, and system with a mixture, ${F}_{M}$ of $Pareto(2.1,1)$ and $Pareto({\alpha}_{2},1)$ service time d.f. vs. the distance between ${F}_{1}$ and ${F}_{M}$, for fixed $p=0.7$ and varying ${\alpha}_{2}=2.1,2.3,\dots ,4.9$.

**Figure 6.**Dependence of the system load, $\rho $, on the parameter ${\alpha}_{2}=2.1,2.3,\dots ,4.9$ of the mixture distribution in an $M/G/c$ system with $c=4$, $\lambda =2.2$, mixture, ${F}_{m}$ of $Pareto(2.1,1)$ and $Pareto({\alpha}_{2},1)$ service time d.f. with mixing coefficient $p=0.7$.

**Figure 7.**Stochastic monotonicity of the system output, in terms of steady-state queue size d.f., on the parameter ${\alpha}_{2}=2.1,2.5,4.9$ of the mixture distribution in an $M/G/c$ system with $c=4$, $\lambda =2.2$, mixture, ${F}_{M}$ of $Pareto(2.1,1)$ and $Pareto({\alpha}_{2},1)$ service time d.f. with mixing coefficient $p=0.7$.

**Figure 8.**Distance between the empirical queue size d.f. in a basic $M/G/c$ system with $c=4$, ${F}_{1}$ being $Pareto({\alpha}_{1},1)$ service time d.f., and system with a mixture, ${F}_{M}$ of $Pareto({\alpha}_{1},1)$ and $Pareto({\alpha}_{2},1)$ service time d.f. vs. the distance between ${F}_{1}$ and ${F}_{M}$, for fixed $p=0.7$, fixed $\rho =0.5$, varying ${\alpha}_{1}=2.1,2.4,\dots ,4.9$ (color), varying ${\alpha}_{2}={\alpha}_{1},{\alpha}_{1}+0.4,\dots ,4.9$ (dot size), and varying $\lambda $, so as to fix the load, $\rho $.

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**MDPI and ACS Style**

Morozov, E.; Pagano, M.; Peshkova, I.; Rumyantsev, A.
Sensitivity Analysis and Simulation of a Multiserver Queueing System with Mixed Service Time Distribution. *Mathematics* **2020**, *8*, 1277.
https://doi.org/10.3390/math8081277

**AMA Style**

Morozov E, Pagano M, Peshkova I, Rumyantsev A.
Sensitivity Analysis and Simulation of a Multiserver Queueing System with Mixed Service Time Distribution. *Mathematics*. 2020; 8(8):1277.
https://doi.org/10.3390/math8081277

**Chicago/Turabian Style**

Morozov, Evsey, Michele Pagano, Irina Peshkova, and Alexander Rumyantsev.
2020. "Sensitivity Analysis and Simulation of a Multiserver Queueing System with Mixed Service Time Distribution" *Mathematics* 8, no. 8: 1277.
https://doi.org/10.3390/math8081277