Question

Two analysts at Exxtol Petrol, Ltd., Log Jam and Bot L. Neck, are having an argument. They are comparing an $\mathrm{M} / \mathrm{M} / 1$ system with mean arrival rate $\lambda$ and mean service rate $2 \mu$ (we assume $\lambda<2 \mu$ ), with an $\mathrm{M} / \mathrm{M} / 2$ system with mean arrival rate $\lambda$ and mean service rate $\mu$ for each server. Log says the $M / M / 2$ system is best because $$ W_{q_{M / M / 2}}<W_{q_{M / M / 1}} . $$ Bot responds that Log has it all wrong because $$ W_{M / M / 1}<W_{M / M / 2}, $$ and therefore, the $\mathrm{M} / \mathrm{M} / 1$ system is superior. Who is right? Note that the two systems have equal capacity; Log and Bot are comparing a single-server system to a double-server system with half-speed servers.

   Two analysts at Exxtol Petrol, Ltd., Log Jam and Bot L. Neck, are having an argument. They are comparing an $\mathrm{M} / \mathrm{M} / 1$ system with mean arrival rate $\lambda$ and mean service rate $2 \mu$ (we assume $\lambda<2 \mu$ ), with an $\mathrm{M} / \mathrm{M} / 2$ system with mean arrival rate $\lambda$ and mean service rate $\mu$ for each server. Log says the $M / M / 2$ system is best because
$$
W_{q_{M / M / 2}}<W_{q_{M / M / 1}} .
$$
Bot responds that Log has it all wrong because
$$
W_{M / M / 1}<W_{M / M / 2},
$$
and therefore, the $\mathrm{M} / \mathrm{M} / 1$ system is superior. Who is right? Note that the two systems have equal capacity; Log and Bot are comparing a single-server system to a double-server system with half-speed servers.
Show more…
Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)
Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)
Arnold O. Allen 2nd Edition
Chapter 5, Problem 19 ↓

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- The $\mathrm{M} / \mathrm{M} / 1$ system has one server with a service rate of $2\mu$ and an arrival rate of $\lambda$. - The $\mathrm{M} / \mathrm{M} / 2$ system has two servers, each with a service rate of $\mu$, and the same arrival rate $\lambda$.  Show more…

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Two analysts at Exxtol Petrol, Ltd., Log Jam and Bot L. Neck, are having an argument. They are comparing an $\mathrm{M} / \mathrm{M} / 1$ system with mean arrival rate $\lambda$ and mean service rate $2 \mu$ (we assume $\lambda<2 \mu$ ), with an $\mathrm{M} / \mathrm{M} / 2$ system with mean arrival rate $\lambda$ and mean service rate $\mu$ for each server. Log says the $M / M / 2$ system is best because $$ W_{q_{M / M / 2}}<W_{q_{M / M / 1}} . $$ Bot responds that Log has it all wrong because $$ W_{M / M / 1}<W_{M / M / 2}, $$ and therefore, the $\mathrm{M} / \mathrm{M} / 1$ system is superior. Who is right? Note that the two systems have equal capacity; Log and Bot are comparing a single-server system to a double-server system with half-speed servers.
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