# On Large Delays in Multi-Server Queues with Heavy Tails

@article{Foss2012OnLD,
title={On Large Delays in Multi-Server Queues with Heavy Tails},
author={Sergey Foss and Dmitry Korshunov},
journal={Math. Oper. Res.},
year={2012},
volume={37},
pages={201-218}
}
• Published 16 April 2011
• Mathematics
• Math. Oper. Res.
We present upper and lower bounds for the tail distribution of the stationary waiting time D in the stable GI/GI/s first-come first-served (FCFS) queue. These bounds depend on the value of the traffic load ρ which is the ratio of mean service and mean interarrival times. For service times with intermediate regularly varying tail distribution the bounds are exact up to a constant, and we are able to establish a “principle of s-k big jumps” in this case (here k is the integer part of ρ), which…
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## References

SHOWING 1-10 OF 24 REFERENCES

### The impact of a heavy-tailed service-time distribution upon the M/GI/s waiting-time distribution

• W. Whitt
• Mathematics
Queueing Syst. Theory Appl.
• 2000
By exploiting an infinite-server-model lower bound, we show that the tails of the steady-state and transient waiting-time distributions in the M/GI/s queue with unlimited waiting room and the

### Waiting Time Asymptotics in the Single Server Queue with Service in Random Order

• Mathematics
Queueing Syst. Theory Appl.
• 2004
It is shown that the waiting time distribution is also regularly varying, with index 1−ν, and the pre-factor is determined explicitly, and another contribution of the paper is the heavy-traffic analysis of the waitingtime distribution in the M/G/1 case.

### Heavy tails in multi-server queues

• Mathematics
• 2013
In this paper, the asymptotic behaviour of the distribution tail of the stationary waiting time $W$ in the $GI/GI/2$ FCFS queue is studied. Under subexponential-type assumptions on the service time

### Heavy Tails in Multi-Server Queue

• Mathematics
Queueing Syst. Theory Appl.
• 2006
Under subexponential-type assumptions on the service time distribution, bounds and sharp asymptotics are given for the distribution tail of the stationary waiting time W in the GI/GI/2 FCFS queue and of a stationary queue length.

### Waiting-Time Asymptotics for the M/G/2 Queue with Heterogeneous Servers

• Mathematics
Queueing Syst. Theory Appl.
• 2002
An exact analysis of the queue length and waiting time distribution in case B(⋅) has a rational Laplace–Stieltjes transform is presented.

### Further delay moment results for FIFO multiserver queues

Using a new family of service disciplines, this work provides weaker sufficient conditions for finite stationary delay moments in FIFO multiserver queues with ρ = E[S]/E[T] <s-1, where S and T are generic service and interarrival times, respectively.

### Fluid Queues with Heavy-Tailed M/G/ Input

• Mathematics, Computer Science
Math. Oper. Res.
• 2005
A fluid queue fed by several heterogeneous M/G/∞ input processes with regularly varying session lengths is considered, and the exact asymptotic behavior of the stationary workload distribution is derived.

### Structural interpretation and derivation of necessary and sufficient conditions for delay moments in FIFO multiserver queues

• Mathematics
Queueing Syst. Theory Appl.
• 2006
This paper derives an alternative derivation of necessary and sufficient conditions for finite stationary delay moments in stable FIFO GI/GI/s queues that incorporate the interaction between service time distribution, traffic intensity and the number of servers in the queue, and provides a structural interpretation of the moment bounds.

### Exact asymptotics for fluid queues fed by multiple heavy-tailed on-off flows

• Mathematics
• 2000
We consider a fluid queue fed by multiple On–Off flows with heavy-tailed (regularly varying) On periods. Under fairly mild assumptions, we prove that the workload distribution is asymptotically

### Sink or swim together: necessary and sufficient conditions for finite moments of workload components in FIFO multiserver queues

• Mathematics
Queueing Syst. Theory Appl.
• 2011
This paper derives moment results for all the components of the stationary workload vector in stable FIFO GI/GI/s queues by exploiting the interaction between service-time distribution, traffic intensity and the number of servers in the queue.