M-M-S Queue

Topics: Queuing Model - Birth-Death Markov Process

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(definition)

An $M/M/S$ queue is a queuing model where:

• The arrivals are determined by a Poisson process (i.e. they follow a Poisson distribution)
• The service times distribute exponentially
• There are $S$ servers

(observation)

Notice that this is basically a birth-death Markov process, since their arrivals also follow a Poisson process and the time between deaths also distributes exponentially.

Arrival and Service Rates §

Tip $\lambda$ and $\mu$

In a queuing model, ${\lambda }_n$ is the expected rate of arrivals per unit of time when there are $n$ clients in the system, while ${\mu }_n$ stands for the expected rate at which clients are being serviced per unit of time when there are $n$ clients in the system.

(theorem)

A $M/M/S$ queue has the following rates:

• If $S=1$, then:
  • ${\lambda }_n=\lambda$ for $n=0,1,2,\dots$
  • ${\mu }_n=\mu$ for $n=1,2,3\dots$
• If $S>1$, then:
  • ${\lambda }_n=\lambda$ for $n=0,1,2,\dots$
  • ${\mu }_n=n\mu$ for $n=1,2,3\dots$
    • Alternatively, ${\mu }_n=S\mu$ for $n=S,S+1,S+2,\dots$

We consider an $M/M/S$ queue stable when its utilisation factor $\rho =\frac{\lambda }{S\mu }$ satisfies $\rho <1$ (i.e. $\lambda <\mu$).

Performance Measures for $M/M/1$ §

(theorem)

When $S=1$, the model is $M/M/1$ and its performance measures are:

• $\rho =\frac{\lambda }{\mu }$ (utilisation factor)
• $C_n={\left(\frac{\lambda }{\mu }\right)}^2={\rho }^n$ for $n=0,1,2,\dots$
• $P_n={\rho }^n(1-\rho )$ for $n=0,1,2,\dots$ (the probability of having $n$ clients in the system)
• $L=\frac{\lambda }{\mu -\lambda }$ (the expected amount of clients in the system)
• $L_q=\rho L=\frac{{\lambda }^2}{\mu (\mu -\lambda )}$ (the expected amount of clients in the queue)
• $W=\frac{1}{\mu -\lambda }$ (the expected wait time in the system)
• $W_q=\rho W=\frac{\lambda }{\mu (\mu -\lambda )}$ (the expected wait time in the queue)

Do note that this assumes that the system is stable (i.e. $\rho <1$).