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Comparison results for M/G/1 queues with waiting and sojourn time deadlines

Part of: Computer system organization Special processes

Published online by Cambridge University Press:  30 July 2019

Yoshiaki Inoue
Yoshiaki Inoue*
Affiliation:
Osaka University
*
*Postal address: Department of Information and Communications Technology, Graduate School of Engineering, Osaka University, Suita 565-0871, Japan.
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Abstract

This paper considers two variants of M/G/1 queues with impatient customers, which are denoted by M/G/1+Gw and M/G/1+Gs. In the M/G/1+Gw queue customers have deadlines for their waiting times, and they leave the system immediately if their services do not start before the expiration of their deadlines. On the other hand, in the M/G/1+Gs queue customers have deadlines for their sojourn times, where customers in service also immediately leave the system when their deadlines expire. In this paper we derive comparison results for performance measures of these models. In particular, we show that if the service time distribution is new better than used in expectation, then the loss probability in the M/G/1+Gs queue is greater than that in the M/G/1+Gw queue.

Keywords

M/G/1 queueimpatient customervirtual waiting timeloss probabilitystochastic order

MSC classification

Primary: 60K25: Queueing theory
Secondary: 68M20: Performance evaluation; queueing; scheduling
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 2 , June 2019 , pp. 524 - 532
Copyright
© Applied Probability Trust 2019 

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References

Baccelli, F., Boyer, P., and Hebuterne, G. (1984). Single-server queues with impatient customers. Adv. Appl. Prob. 16, 887–905.CrossRefGoogle Scholar
Bekker, R., Borst., S. C., and Boxma, O. J. (2004). Queues with workload-dependent arrival and service rates. Queueing Systems 46, 537–556.CrossRefGoogle Scholar
Boxma, O., Perry, D., and Stadje, W. (2011). The M/G/1+G queue revisited. Queueing Systems 67, 207–220.CrossRefGoogle Scholar
Brandt, A. and BRANDT, M. (2013). Workload and busy period for M/G/1 with a general impatience mechanism. Queueing Systems 75, 189–209.CrossRefGoogle Scholar
Brill, P.H. and Posner, M. J. M. (1977). Level crossings in point processes applied to queues: Single- server case. Operat. Res. 25, 662–674.CrossRefGoogle Scholar
Browne, S. and Sigman, K. (1992). Work-modulated queues with applications to storage processes. J. Appl. Prob. 29, 699–712.CrossRefGoogle Scholar
Cohen, J. W. (1977). On up- and downcrossings. J. Appl. Prob. 14, 405–410.CrossRefGoogle Scholar
Daley, D. J. (1965). General customer impatience in the queue GI/G/1. J. Appl. Prob. 2, 186–205.CrossRefGoogle Scholar
Inoue, Y. and Takine, T. (2015). Analysis of the loss probability in the M/G/1+G queue, Queueing Systems 80, 363–386.CrossRefGoogle Scholar
Inoue, Y. and Takine, T. (2015). The M/D/1+D queue has the minimum loss probability among M/G/1+G queues. Operat. Res. Lett. 43, 629–632.CrossRefGoogle Scholar
Kovalenko, I. N. (1961). Some queueing problems with restrictions. Theory Prob. Appl. 6, 205–208.CrossRefGoogle Scholar
Mθller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks, John Wiley, Chichester, UK.Google Scholar
Perry, D. and Asmussen, S. (1995). Rejection rules in the M/G/1 queue. Queueing Systems 19, 105–130.CrossRefGoogle Scholar
Pipkin, A. C. (1991). A Course on Integral Equations, Springer, New York, NY.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders, Springer, New York, NY.CrossRefGoogle Scholar
STANFORD, R. E. (1979). Reneging phenomena in single channel queues. Math. Operat. Res. 4, 162–178.CrossRefGoogle Scholar
Van Houdt, B., Lenin, R.B., and Blondia, C. (2003). Delay distribution of (im)patient customers in a discrete time D-MAP/PH/1 queue with age-dependent service times. Queueing Systems 45, 59–73.CrossRefGoogle Scholar
Van Velthoven, J., Van Houdt, B. and Blondia, C. (2005). Response time distribution in a D-MAP/PH/1 queue with general impatience. Stoch. Models 21, 745–765.CrossRefGoogle Scholar
Van Velthoven, J., Van Houdt, B. and Blondia, C. (2006). On the probability of abandonment in queues with limited sojourn and waiting times. Operat. Res. Lett. 34, 333–338.CrossRefGoogle Scholar