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The P-K formulas for our M/M/1 queueing system also come in handy here:
So:
What is the average fraction of time the channel is busy, and what is the average number of frames in service?
This is a trick question. Both of these are different ways to ask for the utilization of the system. The utilization was given to us as 0.6, thus the answer to both these questions is 0.6.
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Since we are modeling a real-world, physical system with a single server that works in units of frames, either a single frame is being serviced or no single frame is being serviced. Thus, the answer is no; there must be either 0 or 1 frames in service at any given instant. ρ, on the other hand, denotes the probability that a frame is being serviced at an arbitrary moment in time.
Example 2
Consider a single server FCFS queueing system with interarrival times that are exponentially distributed with a mean value of 3 minutes and service times that are exponentially distributed with a mean value of 2 minutes.
Given this little bit of information, we can deduce that the arrival process is Poisson with arrival rate λ = 1/3 customers/minute, the service rate is μ = 1/2 customers/minute, and that the queueing system is M/M/1.
What is the probability that an arrival will have to wait for service?
This is simply the probability that an arrival finds the server busy, which is the utilization. Thus, we can compute the utilization, ρ, as:
What is the average length of the queue?
We can directly use the P-K formulas for an M/M/1 queueing system, and compute the average number of customers waiting in line as:
If the mean service time is 2 minutes, what must the mean interarrival time be so that the average wait in the queue is 5 minutes?
We want to find 1/λ (the average interarrival time), and we know the average service time is 1/μ = 2 minutes. The utilization, ρ, is ρ = λ/μ, so we have a way of finding 1/λ given that we can determine the utilization.
As given by the P-K formulas for an M/M/1 queueing system, we can compute the utilization using the known values of μ and W:
From this, we can directly compute the average interarrival time using the definition of utilization:
For the mean interarrival time computed above, find the average transaction time and the average number of waiting customers in the system
The average transaction time is the time a customer spends in the system, which is the wait time in the queue plus the service time:
We can use Little's Law directly to compute the average number of customers in the system:
At what mean interarrival time does the queue grow arbitrarily large?
If the queue were to grow arbitrarily large, the system would be considered unstable given the fixed service rate. Thus, we want to find the point at which the system becomes unstable. Stability is defined as the utilization being less than or equal to 1:
Thus, we can solve for 1/λ, which is the average interarrival time:
Example 3
Example 4
Example 5
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