# Asymptotic behaviour of the tandem queueing system with identical service times at both queues

@article{Boxma2000AsymptoticBO, title={Asymptotic behaviour of the tandem queueing system with identical service times at both queues}, author={Onno J. Boxma and Q. Deng}, journal={Mathematical Methods of Operations Research}, year={2000}, volume={52}, pages={307-323} }

Abstract. Consider a tandem queue consisting of two single-server queues in series, with a Poisson arrival process at the first queue and arbitrarily distributed service times, which for any customer are identical in both queues. For this tandem queue, we relate the tail behaviour of the sojourn time distribution and the workload distribution at the second queue to that of the (residual) service time distribution. As a by-result, we prove that both the sojourn time distribution and the workload…

## 11 Citations

### Heavy traffic limit for a tandem queue with identical service times

- Mathematics, BusinessQueueing Syst. Theory Appl.
- 2018

A two-node tandem queueing network in which the upstream queue is M/G/1 and each job reuses its upstream service requirement when moving to the downstream queue is considered, in which both servers employ the first-in-first-out policy.

### Steady-state asymptotics for tandem, split-match and other feedforward queues with heavy tailed service

- MathematicsQueueing Syst. Theory Appl.
- 1999

It is shown that when the service time at the first node has a lighter tail, the tail of steady-state delay at the second node, D(2), has the same asymptotics as if it were a GI/GI/1 queue, and for split-match (fork-join) queues with subexponential service times, the same result holds.

### Delays in a series of queues: Independent versus identical service times

- Business, Computer ScienceThe IEEE symposium on Computers and Communications
- 2010

It turns out that in light traffic the delays in case of identical service times are larger than for independent service times by a factor that first increases with increasing traffic intensity up to a maximum value approached in medium traffic after which it decreases quickly and drops down to become significantly smaller than one in heavy traffic.

### Delays in a series of queues with correlated service times

- Computer Science, BusinessJ. Netw. Comput. Appl.
- 2012

### Tails for (max, plus) recursions under subexponentiality

- Mathematics, Computer ScienceQueueing Syst. Theory Appl.
- 2006

Under subexponentiality assumptions on the input to the recursion, the tail asymptotics of certain (max, plus)-linear functionals of this solution arise from FIFO queueing networks; more specifically, from stochastic event graphs.

### About the Sojourn Time Process in Multiphase Queueing Systems

- Mathematics
- 2006

Multiphase queueing systems (MQS) (tandem queues, queues in series) are of special interest both in theory and in practical applications (packet switch structures, cellular mobile networks, message…

### Moments and tails in monotone-separable stochastic networks

- Mathematics, Computer Science
- 2004

The purpose is the analysis of the tails of the stationary state variables in the particular case of i.i.d. driving sequences and establishes general comparison relationships between networks of this class and the GI/GI/1/\\infty queue.

### A Law of the Iterated Logarithm for the Sojourn Time Process in Queues in Series

- Mathematics
- 2016

The object of this research in the sphere of queueing theory is the law of the iterated logarithm under the conditions of heavy traffic in queues in series. In this paper, the laws of the iterated…

### Queues with regular variation

- Materials Science
- 2001

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of…

## References

SHOWING 1-10 OF 21 REFERENCES

### On a tandem queueing model with identical service times at both counters, II

- MathematicsAdvances in Applied Probability
- 1979

This paper is devoted to the practical implications of the theoretical results obtained in Part I [1] for queueing systems consisting of two single-server queues in series in which the service times…

### The M/G/1 queue with heavy-tailed service time distribution

- MathematicsIEEE J. Sel. Areas Commun.
- 1998

An approximation for W(t), which is based on a heavy-traffic limit theorem for the M/G/1 queue with heavy-tailed service time distribution (with infinite variance), is presented and shown to yield excellent results for values of t which are not too small, even when the load is not heavy.

### Heavy-traffic analysis of the M/G/1 queue with priority classes

- Mathematics
- 1998

It is shown for this heavy-tailed case that the waiting time distribution of the low-priority customers is regularly varying of index one degree higher than that of the service time distribution with the heaviest tail.

### Networks of Queues with Long-Range Dependent Traffic Streams

- Mathematics
- 1996

Traditional applications of queuing theory in the analysis and design of communication networks often use the fact that quasi-reversible queues can be interconnected with Bernoulli routing to form network models whose stationary distribution is of product form, but recent empirical studies have provided strong evidence that the traffic to be carried by the next generation of communication Networks exhibits long-range dependence, implying that it cannot be satisfactorily modeled as a Poisson process.

### Self-Similar Network Traffic and Performance Evaluation

- Computer Science
- 2000

Self-Similar Network Traffic: An Overview (K. Park & W. Willinger). Wavelets for the Analysis, Estimation, and Synthesis of Scaling Data (P. Abry, et al.). Simulations with Heavy-Tailed Workloads (M.…

### The longest service time in a busy period

- Computer ScienceZ. Oper. Research
- 1980

The supremumm of the service times of the customers served in a busy period of the G/G/s queue is considered and the tail behaviour of the distributionm (w) ofm is compared with that of theservice time distribution.

### On the tails of waiting-time distributions

- MathematicsJournal of Applied Probability
- 1975

Results are given which relate the tail behaviour of the service and limiting waiting time distributions of a GI/G/1 queue. A limit theorem for the maxima of waiting times is given.

### Subexponential distributions and integrated tails

- MathematicsJournal of Applied Probability
- 1988

Let F be a distribution function on [0,∞) with finite expectation. In terms of the hazard rate of F several conditions are given which simultaneously imply subexponentiality of F and of its…

### Asymptotics of stochastic networks with subexponential service times

- MathematicsQueueing Syst. Theory Appl.
- 1999

The asymptotics of stationary response times in a class of stochastic Petri nets called event graphs are given, based on the assumption that the firing times are independent and that the tail of the firing time of one of the transitions is subexponential and heavier than those of the others.

### Makarichev Analysis of a tandem queueing system with identical service times at both counters for various service disciplines

- Proc. 3rd ITC Seminar