Tandem queue with server slow-down

ABSTRACT

We study how rare events happen in the standard two-node tandem Jackson queue and in a generalization, the socalled slow-down network, see [2]. In the latter model the service rate of the first server depends on the number of jobs in the second queue: the first server slows down if the amount of jobs in the second queue is above some threshold and returns to its normal speed when the number of jobs in the second queue is below the threshold. This property protects the second queue, which has a finite capacity B, from overflow. In fact this type of overflow is precisely the rare event we are interested in. More precisely, consider the probability of overflow in the second queue before the entire system becomes empty. The starting position of the two queues may be any state in which at least one job is present.

REFERENCES

• 1: V. Anantharam, P. Heidelberger and P. Tsoucas (1990). Analysis of rare events in continuous time Markov chains via time reversal and fluid approximation. IBM Research Report RC 16280.
• 2: N. D. van Foreest, M. R. H. Mandjes, J. C. W. van Ommeren, W. R. W. Scheinhardt (2005). A tandem queue with server slow-down and blocking. Stochastic Models 21 (2--3), 695--724.
• 3: Paul Glasserman , Shing-Gang Kou, Analysis of an importance sampling estimator for tandem queues, ACM Transactions on Modeling and Computer Simulation (TOMACS), v.5 n.1, p.22-42, Jan. 1995 [doi>10.1145/203091.203093]
• 4: Dirk P. Kroese , Victor F. Nicola, Efficient simulation of a tandem Jackson network, ACM Transactions on Modeling and Computer Simulation (TOMACS), v.12 n.2, p.119-141, April 2002 [doi>10.1145/566392.566395]
• 5: D.I. Miretskiy , W.R.W. Scheinhardt , M.R.H. Mandjes, Efficient Simulation of a Tandem Queue with Server Slow-down, Simulation, v.83 n.11, p.751-767, November 2007 [doi>10.1177/0037549707086193]
• 6: S. Parekh, J. Walrand (1989). A quick simulation method for excessive backlogs in networks of queues. IEEE Transactions on Automatic Control 34, 54--66.

INDEX TERMS

Primary Classification:
  D. Software
  D.4 OPERATING SYSTEMS
      D.4.7 Organization and Design
          Subjects: Distributed systems

Additional Classification:
  D. Software
  D.4 OPERATING SYSTEMS
      D.4.4 Communications Management
      D.4.8 Performance
          Subjects: Queueing theory; Operational analysis

General Terms:
Algorithms, Design, Management, Performance