Comparisons of Exhaustive and Nonexhaustive M/M/1/N Queues with Working Vacation and Threshold Policy

  • Wei Sun
  • Shiyong LiEmail author
  • Yan Wang
  • Naishuo Tian


This paper compares the performance of exhaustive and nonexhaustive M/M/1/N queues with working vacation and threshold policy. In an exhaustive queue, the server slows down its service rate only when no customers exist in the system, and turns to normal service until the number of customers achieves a threshold. However, in a nonexhaustive queue, the server switches service rate between a low and a high value depending on system congestion. To get equilibrium arrival rate of customers and social welfare for the two types of queues, we first derive queue length distributions and expected busy circle. Then, by making sensitivity analysis of busy circle, system cost, arrival rate and optimal social welfare, we find that customers tend to join exhaustive queues instead of nonexhaustive queues, and the optimal threshold in an exhaustive queue is probably inconsistent with the one in a nonexhaustive queue. Moreover, in general, whether to consider system cost or not in social welfare will obviously affect the tendencies of optimal arrival rate and optimal social welfare with the threshold and system capacity for the two types of queues, especially for the nonexhaustive queues, and then affect the final decisions of social planner or system manager.


Markovian queue service discipline limited capacity working vacations threshold policy equilibrium arrival rate busy circle social welfare system cost 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors would like to thank the anonymous reviewers for the useful comments on this work, and the support from the National Natural Science Foundation of China under Grant 71671159, the Humanity and Social Science Foundation of Ministry of Education of China under Grant 16YJC630106, the Natural Science Foundation of Hebei Province under Grants G2016203236 and G2018203302, and the project Funded by Hebei Education Department under Grants BJ2016063 and BJ2017029, and Hebei Talents Program under Grant A2017002108.


  1. Abbas K, Heidergott B, Aissani D (2013). A functional approximation for the M/G/1/N queue. Discrete Event Dynamic Systems-Theory and Applications 23(1): 93–104.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Akar N (2012). Moments of conditional sojourn times in finite capacityM/M/1/N−PS processor sharing queues. IEEE Communications Letters 16(4): 533–535.CrossRefGoogle Scholar
  3. Banik A, Gupta U, Pathak S (2007). On the GI/M/1/N queue with multiple working vacations-analytic analysis and computation. Applied Mathematical Modelling 31(9): 1701–1710.CrossRefzbMATHGoogle Scholar
  4. Dimitrakopoulos Y, Burnetas A (2011). Customer equilibrium and optimal strategies in an M/M/1 queue with dynamic service control. Working paper, University of Athens, Greece.zbMATHGoogle Scholar
  5. Guo P, Hassin R (2011). Strategic behavior and social optimization in Markovian vacation queues. Operations Research 59(4): 986–997.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Guo P, Hassin R (2012). Strategic behavior and social optimization in Markovian vacation queues: The case of heterogeneous customers. European Journal of Operational Research 222(2): 278–286.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Guo P, Li Q (2013). Strategic behavior and social optimization in partially-observable Markovian vacation queues. Operations Research Letters 41: 277–284.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Ke J, Wu C, Zhang ZG (2010). Recent developments in vacation queueing models: Ashort survey. International Journal of Operations Research 7: 3–8.Google Scholar
  9. Li J, Tian N, Zhang ZG, Luh H (2009). Analysis of the M/G/1 queue with exponentially working vacations–A matrix analytic approach. Queueing Systems 61(2–3): 139–166.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Ouazine S, Abbas K (2016). Development of computational algorithmfor multiserver queue with renewal input and synchronous vacation. Applied Mathematical Modelling 40(2): 1137–1156.MathSciNetCrossRefGoogle Scholar
  11. Servi L, Finn S (2002). M/M/1 queues with working vacations (M/M/1/N/WV). Performance Evaluation 50: 41–52.CrossRefGoogle Scholar
  12. Sun W, Li S (2014). Equilibrium and optimal behavior of customers in Markovian queues with multiple working vacations. TOP 22(2): 694–715.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Sun W, Li S (2016). Equilibrium and optimal balking strategies of customers in Markovian queues with multiple vacations and N-policy. Applied Mathematical Modelling 40(1): 284–301.MathSciNetCrossRefGoogle Scholar
  14. Sun W, Li S, Li Q (2014). Equilibrium balking strategies of customers in Markovian queues with two-stage working vacations. Applied Mathematics and Computation 248: 195–214.CrossRefzbMATHGoogle Scholar
  15. Tian N, Li J, Zhang ZG (2009). Matrix analytic method and working vacation queues–A survey. International Journal of Information and Management Sciences 20: 603–633.MathSciNetzbMATHGoogle Scholar
  16. Wang F, Wang J, Zhang F (2014). Equilibrium customer strategies in the Geo/Geo/1 queue with single working vacation. Discrete Dynamic in Nature and Society, No.309489.Google Scholar
  17. Zhang F, Wang J, Liu B (2013). Equilibrium balking strategies in Markovian queues with working vacations. Applied Mathematical Modelling 37(16–17): 8264–8282.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Systems Engineering Society of China and Springer-Verlag GmbH Germany 2019

Authors and Affiliations

  1. 1.School of Economics and ManagementYanshan UniversityQinhuangdaoChina
  2. 2.College of ScienceYanshan UniversityQinhuangdaoChina

Personalised recommendations