7.5 Steady-State Behavior of the M/M/1 Model

Henceforth, we assume that the calling population is infinite, the arrivals are assumed to follow a Poisson process with rate \(\lambda\) arrivals per time unit - that is, the inter-arrival times are assumed to be exponentially distributed with mean \(1/\lambda\). We also assume that service times are exponentially distributed with mean \(1/\mu\). The queue discipline will be FIFO. Because of the exponential distribution assumption on the arrival process, these models are called Markovian.

A queuing system is said to be in statistical equilibrium or in steady state if the probability that the system is in a given state is not time-dependent: that is \[ P(L(t)=n)= P_n(t) = P_n \] is independent of time \(t\). Two properties of such systems:

  • they approach statistical equilibrium from any starting state,

  • they remain in statistical equilibrium once it is reached.

For the models that we will consider next the steady-state parameter \(L\), the average number of customers in the system, can be computed as \[ L= \sum_{n=0}^\infty nP_n \] where \(\{P_0,P_1,\dots\}\) are the steady-state probabilities of finding \(n\) customers in the system. Once \(L\) is given, the other steady-state parameters can be computed easily from Little’s law: \[\begin{align*} w &= \frac{L}{\lambda}\\ w_Q &= w - \frac{1}{\mu}\\ L_Q &= \lambda w_Q, \end{align*}\] where \(\lambda\) is the arrival rate and \(\mu\) is the service rate.

Recall that for queues to reach statistical equilibrium a necessary and sufficient condition is that $/c<1 $.

For the M/M/1 model the steady-state parameters can be shown to be equal to: \[\begin{align*} \rho &= \frac{\lambda}{\mu} \\ L &= \frac{\lambda}{\mu-\lambda}\\ w &= \frac{1}{\mu-\lambda}\\ w_Q &= \frac{\lambda}{\mu(\mu-\lambda)}\\ L_Q &= \frac{\lambda^2}{\mu(\mu-\lambda)}\\ P_n &= \left(1-\frac{\lambda}{\mu}\right)\left(\frac{\lambda}{\mu}\right)^n \end{align*}\]

Let’s consider an example. Suppose that the interarrival times and service times of a single-chair unisex hair-styling shop have been shown to be exponentially distributed. The values of \(\lambda\) and \(\mu\) are 2 per hour and 3 per hour, respectively: that is, the time between arrivals averages 1/2 hour and the service time averages 20 minutes. The server utilization and the probabilities for 0, 1, 2, 3, and 4 or more customers in the shop are computed as follows: \[\begin{align*} \rho &= \frac{\lambda}{\mu}=2/3\\ P_0 &= 1- \frac{\lambda}{\mu} = 1/3\\ P_1 &= \left(\frac{1}{3}\right) \left(\frac{2}{3}\right) =\frac{2}{9}\\ P_2 &= \left(\frac{1}{3}\right) \left(\frac{2}{3}\right)^2 =\frac{4}{27}\\ P_3 &= \left(\frac{1}{3}\right) \left(\frac{2}{3}\right)^3 =\frac{8}{81}\\ P_{\geq 4}&= 1-\sum_{n=0}^3=\frac{16}{81} \end{align*}\] From the above calculations the probability that the hair stylist is busy is \(1-P_0=0.67\) and thus the probability that he is free is \(0.33\). The average number of customer in the system is \[ L=\frac{\lambda}{\mu-\lambda}=\frac{2}{3-2}= 2 \mbox{ customers} \] The average time in the system is \[ w = \frac{L}{\lambda}=\frac{1}{\mu-\lambda} = \frac{2}{2}=1 \mbox{ hour} \] The average time the customer spends in the queue is \[ w_Q = w - \frac{1}{\mu}= 1- \frac{1}{3}= \frac{2}{3} \mbox{ hour} \] and the number of customer in the queue is \[ L_Q = \frac{\lambda^2}{\mu(\mu-\lambda)}=\frac{4}{3} \mbox{ customers} \]

Let’s consider another example. Arrivals occur at rate \(\lambda = 10\) per hour and management has a choice of two servers, one who works at rate \(\mu_1=11\) customers per hour and the second at rate \(\mu_2 = 12\) customers per hour. The utilizations are \(\rho_1 = 10/11=0.909\) and \(\rho_2 = 10/12 = 0.833\). Assume a M/M/1 model, then \[ L_1 = \frac{\lambda}{\mu_1-\lambda}=\frac{10}{11-10}=10 \] whilst \[ L_2 = \frac{\lambda}{\mu_2-\lambda}=\frac{10}{12-10}=5 \] Thus an increase in service rate from 11 to 12 customers per hour, a mere 9.1% increase, would result in a decrease in average number in system from 10 to 5, which is a 50% decrease!