M/G/k queue: Difference between revisions

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==Delay/waiting time distribution==

ApproximationsThere existare numerous approximations for the average delay a job experiences.<ref>{{cite name="tijms"jstor|169760‎}}</ref><ref>{{cite jstor|1426432}}</ref><ref>{{cite Itjstor|3212698‎}}</ref><ref>{{cite isjstor|3213437‎}}</ref><ref>{{cite knownjstor|172087‎}}</ref><ref>{{cite thatjstor|170637‎}}</ref> noThe approximationfirst such was given in 1959 using onlya thefactor firstto twoadjust momentsthe canmean bewaiting accuratetime in allan cases.[[M/M/c queue]]<ref name="gbdz" /><ref>{{cite doi|10.10071057/s11134-009-9133-xjors.1959.5}}</ref>

:<math>E[W^{\text{M/G/}k}] = \frac{C^2+1}{2} \mathbb E [ W^{\text{M/M/}c}]</math>

where ''C''² is the [[coefficient of variation]] of the service time distribution. [[Ward Whitt]] described this approximation as “usually an excellent approximation, even given extra information about the service-time distribution."<ref>{{cite doi|10.1111/j.1937-5956.1993.tb00094.x}}</ref>

However, it is known that no approximation using only the first two moments can be accurate in all cases.<ref name="gbdz">{{cite doi|10.1007/s11134-009-9133-x}}</ref>

A [[Markov–Krein]] characterisation has been shown to produce tight bounds.<ref>{{cite doi|10.1007/s11134-011-9248-8}}</ref>