Real Time Systems Design And Analysis

Chapter 7.3.1 - The M/M/1 Queue

7.3.1   The M/M/1 Queue

The simplest queuing model is the M/M/1 queue, which represents a single-
server system with a Poisson arrival model (exponential interarrival times for
the customers or interrupt requests with mean 1/λ), and exponential service or
process time with mean 1/μ and λ < μ . As suggested before, this model can be
used effectively to model certain aspects of real-time systems; it is also useful
because it is well known, and several important results are immediately available
[Kleinrock75]. For example, let N be the number of customers in the queue.
Letting ρ = λ/μ, then the average number of customers in the queue in such a
system is

 

with variance

 

The average time a customer spends in the system is

 

The random variable Y for the time spent in the system has probability distribution

 

with y ≥ 0.

Finally, it can be shown that the probability that at least k customers are in
the queue is

 

In the M/M/1 model, the probability of exceeding a certain number of customers
in the system decreases geometrically. If interrupt requests are considered customers
in a certain system, then two such requests in the system at the same time
(a time-overloaded condition) have a far greater probability of occurrence than
three or more such requests. Thus, building systems that can tolerate a single
time-overload will contribute significantly to system reliability, while worrying
about multiple time-overload conditions is probably futile. The following sections
describe how the M/M/1 queue can be used in the analysis of real-time systems.

 

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