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The M/M/m queue can be used to model multiprocessor systems or devices that have several identical servers and all jobs waiting for these servers are kept in one queue. It is assumed that there are m servers each with a service rate of µ jobs per unit time. The arrival rate is λ jobs per unit time. If any of the m servers are idle, the arriving job is serviced immediately. If all m servers are busy, the arriving jobs wait in a queue. The state of the system is represented by the number of jobs n in the system. The state transition diagram is shown in Figure 31.4. It is easy to see that the number of jobs in the system is a birth-death process with the following correspondence:
Theorem 31.1 gives us the following expression for the probability of n jobs in the system:
In terms of the traffic intensity ρ = ρ/mµ, we have
Using this expression for pn, other performance parameters for M/M/m queues can be easily derived. A summary of the results is presented in Box 31.2. Derivations of some of the results are given here. Readers who are not interested in the derivation may sldp to the example following it.
FIGURE 31.4 State transition diagram for an M/M/m queue.
Box 31.2 M/M/m Queue
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F(w) = 1 emµ(1ρ)w
The center can be modeled as an M/M/5 queueing system with an arrival rate of per minute and a service rate of
per minute.
Substituting these into the expressions previously used in this section, we get
The probability of all terminals being idle is
The probability of all terminals being busy is
Average terminal utilization is
ρ = 0.67
Average number of students in the center is
The average number of students waiting in the queue is
The average number of students using the terminals is
The mean and variance of the time spent in the center are
Thus, each student spends an average of 24 minutes in the center. Of these, 20 minutes are spent working on the terminal and 4 minutes are spent waiting in the queue. We can further verify this using the formula for the mean waiting time:
The 90-percentile of the waiting time is
Thus, 10% of the students have to wait more than 14 minutes.
Queueing models can be used not only to study the current behavior but also to predict what would happen if we made changes to the system. The following examples illustrate this.
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