Mathematical Methods of Operations Research

, Volume 73, Issue 1, pp 139–152 | Cite as

The age of the arrival process in the G/M/1 and M/G/1 queues

  • Moshe Haviv
  • Yoav KernerEmail author
Original Article


This paper shows that in the G/M/1 queueing model, conditioning on a busy server, the age of the inter-arrival time and the number of customers in the queue are independent. The same is the case when the age is replaced by the residual inter-arrival time or by its total value. Explicit expressions for the conditional density functions, as well as some stochastic orders, in all three cases are given. Moreover, we show that this independence property, which we prove by elementary arguments, also leads to an alternative proof for the fact that given a busy server, the number of customers in the queue follows a geometric distribution. We conclude with a derivation for the Laplace Stieltjes Transform (LST) of the age of the inter-arrival time in the M/G/1 queue.


G/M/1 queue M/G/1 queue Age of inter arrival time 


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  1. Adan I, Haviv M (2009) Conditional ages and residual service times in the M/G/1 queue. Stoch Models 25: 110–128zbMATHCrossRefMathSciNetGoogle Scholar
  2. Adan I, Boxma O, Perry D (2005) The G/M/1 queue revisited. Math Methods Oper Res 62: 437–452zbMATHCrossRefMathSciNetGoogle Scholar
  3. Altman E, Jimenez T, Nunez-Queija R, Yechiali U (2004) Optimal routing among ·/M/1 queues with partial information. Stoch Models 20: 149–172zbMATHCrossRefMathSciNetGoogle Scholar
  4. Cohen JW (1982) The single server queue, 2nd edn. North-Holland, AmsterdamzbMATHGoogle Scholar
  5. Fakinos D (1990) Equilibrium queue size distributions for semi-reversible queues. Stoch Processes Appl 36: 331–337zbMATHCrossRefMathSciNetGoogle Scholar
  6. Haviv M, van der Heyden L (1984) Perturbation bounds for the stationary probabilities of a finite Markov chain. Adv Appl Probab 16: 804–818zbMATHCrossRefGoogle Scholar
  7. Haviv M, van Houtum G-J (1998) The critical offered load in variants of the symmetric shortest and longest queue systems. Stoch Models 14: 1179–1195zbMATHCrossRefGoogle Scholar
  8. Haviv M, Zlotnikov R (2009) Computational schemes for two exponential servers where the first has a finite buffer. Working paper. Currently available at
  9. Kerner Y (2008) On the joint distribution of queue length and residual service time in the M n/G/1 queue. Stoch Models 24: 364–375zbMATHCrossRefMathSciNetGoogle Scholar
  10. Nunez-Queija R (2001) Note on the GI/GI/1 queue with LCFS-PR observed at arbitrary times. Probab Eng Inf Sci 15: 179–187zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of StatisticsThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Department of Industrial Engineering and ManagementBen-Gurion University of the NegevBeer ShevaIsrael

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