M/M/1 Queueing System

We have already covered queueing theory basics in a previous article. In this article we will focus on M/M/1 queueing system. As we have seen earlier, M/M/1 refers to negative exponential arrivals and service times with a single server. This is the most widely used queueing system in analysis as pretty much everything is known about it. M/M/1 is a good approximation for a large number of queueing systems.

The following topics will be discussed in detail:

• Poisson Arrivals
• Poisson Service Times
• Single Server
• M/M/1 Results

Poisson Arrivals

M/M/1 queueing systems assume a Poisson arrival process. This assumption is a very good approximation for arrival process in real systems that meet the following rules:

1. The number of customers in the system is very large.
2. Impact of a single customer on the performance of the system is very small, i.e. a single customer consumes a very small percentage of the system resources.
3. All customers are independent, i.e. their decision to use the system are independent of other users.

Cars on a Highway

As you can see these assumptions are fairly general, so they apply to a large variety of systems. Lets consider the example of cars entering a highway. Lets see if the above rules are met.

1. Total number of cars driving on the highway is very large.
2. A single car uses a very small percentage of the highway resources.
3. Decision to enter the highway is independently made by each car driver.

The above observations mean that assuming a Poisson arrival process will be a good approximation of the car arrivals on the highway. If any one of the three conditions is not met, we cannot assume Poisson arrivals. For example, if a car rally is being conducted on a highway, we cannot assume that each car driver is independent of each other. In this case all cars had a common reason to enter the highway (start of the race).

Telephony Arrivals

Lets take another example. Consider arrival of telephone calls to a telephone exchange. Putting our rules to test we find:

1. Total number of customers that are served by a telephone exchange is very large.
2. A single telephone call takes a very small fraction of the systems resources.
3. Decision to make a telephone call is independently made by each customer.

Again, if all the rules are not met, we cannot assume telephone arrivals are Poisson. If the telephone exchange is a PABX catering to a few subscribers, the total number of customers is small, thus we cannot assume that rule 1 and 2 apply. If rule 1 and 2 do apply but telephone calls are being initiated due to some disaster, calls cannot be considered independent of each other. This violates rule 3.

Poisson Arrival Process

Now that we have established scenarios where we can assume an arrival process to be Poisson. Lets look at the probability density distribution for a Poisson process. This equation describes the probability of seeing n arrivals in a period from 0 to t.

Where:

• t is used to define the interval 0 to t
• n is the total number of arrivals in the interval 0 to t.
• lambda is the total average arrival rate in arrivals/sec.

Negative Exponential Arrivals

We have seen the Poisson probability distribution. This equation gives information about how the probability is distributed over a time interval. Unfortunately it does not give an intuitive feel of this distribution. To get a good grasp of the equation we will analyze a special case of the distribution, the probability of no arrivals taking place over a given interval. 

Its easy to see that by substituting n with 0, we get the following equation:

This equation shows that probability that no arrival takes place during an interval from 0 to t is negative exponentially related to the length of the interval. This is better illustrated with an example.