# On the inapproximability of M/G/K: why two moments of job size distribution are not enough

@article{Gupta2010OnTI,
title={On the inapproximability of M/G/K: why two moments of job size distribution are not enough},
author={Varun Gupta and Mor Harchol-Balter and Jim G. Dai and Bert Zwart},
journal={Queueing Systems},
year={2010},
volume={64},
pages={5-48}
}
• Published 2010
• Mathematics
• Queueing Systems
The M/G/K queueing system is one of the oldest models for multiserver systems and has been the topic of performance papers for almost half a century. However, even now, only coarse approximations exist for its mean waiting time. All the closed-form (nonnumerical) approximations in the literature are based on (at most) the first two moments of the job size distribution. In this paper we prove that no approximation based on only the first two moments can be accurate for all job size distributions…
On Markov-Krein Characterization of Mean Sojourn Time in Queueing Systems
• Mathematics
• 2011
We present a new analytical tool for three queueing systems which have defied exact analysis so far: (i) the classical M/G/k multi-server system, (ii) queueing systems with fluctuating arrival and
Simple and explicit bounds for multi-server queues with universal 1 / (1 - rho) scaling
• Mathematics, Computer Science
• 2017
We consider the FCFS GI/GI/n queue, and prove the first simple and explicit bounds that scale gracefully and universally as 1 / (1 - rho) (and better), with rho the corresponding traffic intensity.
Breaking the dimensionality curse in multi-server queues
• Computer Science
Comput. Oper. Res.
• 2016
Simple and explicit bounds for multi-server queues with $1/(1 - \rho)$ (and sometimes better) scaling
• Mathematics
• 2017
We consider the FCFS GI/GI/n queue, and prove the first simple and explicit bounds that scale as 1 1−ρ (and sometimes better). Here ρ denotes the corresponding traffic intensity. Conceptually, our
Reduced complexity in M/Ph/c/N queues
• Computer Science
Perform. Evaluation
• 2014
Extremal GI/GI/1 Queues Given Two Moments: Three-Point and Two-Point Distributions
This paper exposes some important open problems in queueing theory. We use simulation and optimization to evaluate the tight upper and lower bounds for the transient and steady-state mean waiting
On Markov–Krein characterization of the mean waiting time in M/G/K and other queueing systems
• Mathematics
Queueing Syst. Theory Appl.
• 2011
An analogous to the classical Markov–Krein theorem, it is conjecture that the bounds on the mean waiting time are achieved by service distributions corresponding to the upper/lower principal representations of the moment sequence.
Set-Valued Performance of Queues Given Partial Information
In order to understand queueing performance given only partial information about the model, we propose determining the set of all possible values given that limited information. We illustrate this
SET-VALUED PERFORMANCE APPROXIMATIONS FOR THE $GI/GI/K$ QUEUE GIVEN PARTIAL INFORMATION
• Mathematics
Probability in the Engineering and Informational Sciences
• 2022
This work designs extremal models yielding tight upper and lower bounds on the asymptotic decay rate of the steady-state waiting-time tail probability and illustrates the theoretically justified intervals of values for the decay rate and the associated heuristically determined interval ofvalues for the mean waiting times.
Perfect simulation of M/G/c queues
• Mathematics
Advances in Applied Probability
• 2015
A perfect simulation algorithm for the stable M/G/c queue is described and a careful analysis shows that appropriate ordering relationships can still be maintained, so long as service durations continue to be coupled in order of initiation of service.

## References

SHOWING 1-10 OF 67 REFERENCES
New bounds for expected delay in FIFO M/G/c queues
• Mathematics, Computer Science
Queueing Syst. Theory Appl.
• 1997
This paper exploits a new delay recursion for the GI/GI/c queue to produce bounds of a different sort when the traffic intensity p = λ/μ = E[S]/E[T] is less than the integer portion of the number of servers divided by two.
Approximations of the Mean Waiting Time in an M/G/s Queueing System
• Mathematics
Oper. Res.
• 1979
In order to obtain mean waiting time approximations it appears to be useful to introduce a quantity (the “normed cooperation coefficient”) which is inversely proportional to WGs andWhich is in some sense a measure for the “cooperation” between the servers of the service facility.
The effect of variability in the GI/G/s queue
• W. Whitt
• Mathematics
Journal of Applied Probability
• 1980
In 1969 H. and D. Stoyan showed that the stationary waiting-time distribution in a GI/G/1 queue increases in the ordering determined by the expected value of all non-decreasing convex functions when
A semidefinite optimization approach to the steady-state analysis of queueing systems
• Computer Science, Mathematics
Queueing Syst. Theory Appl.
• 2007
An approach based on semidefinite optimization to find bounds on moments of the waiting time under moment information on the service and interarrival times is proposed and it is shown that the classical Kingman's and Daley’s bounds for the expected waiting time in a GI/GI/1 queue are special cases of the proposed approach.
Approximation of the Mean Queue Length of an M/G/c Queueing System
• Mathematics, Computer Science
Oper. Res.
• 1995
Numerical results show that the approximation is accurate even when the coefficient of variation of the service time and the number of channels of the system are as large as 20 and 200, respectively.
Some comparability results for waiting times in single- and many-server queues
• Mathematics
• 1984
It is shown that the stationary waiting time random variables W′, W″ of two M/G/l queueing systems for which the corresponding service time random variables satisfy E(S ′−x)+ ≦ E(S ″−x)+ (all x >0),
A Diffusion Approximation for the G/GI/n/m Queue
A diffusion approximation for the queue-length stochastic process in the G/GI/n/m queueing model (having a general arrival process, independent and identically distributed service times with a general distribution,n servers, andm extra waiting spaces) is developed, focusing especially upon the steady-state delay probability.
On approximations for queues, I: Extremal distributions
• W. Whitt
• Mathematics, Computer Science
AT&T Bell Laboratories Technical Journal
• 1984
This work calculates the set of possible values for the mean queue length in a GI/M/1 queue and shows how it depends on the traffic intensity and the second moment, and uses extremal distributions to compare alternative parameters for approximations.
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.
APPROXIMATIONS FOR THE GI/G/m QUEUE
Approximations for a basic queueing model, which has m identical servers in parallel, unlimited waiting room, and the first-come first-served queue discipline, are developed and evaluated and are useful supplements to algorithms for computing the exact values that have been developed in recent years.