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M/M/s Queuing Optimization Spreadsheet is now available only at $9.99
M/M/1 Queuing System
Now we will go into the detail of the performance for a special simple queuing system where there is only single server and the arrival distribution of customers follows Poisson distribution and the distribution for service time follows Exponential distribution.
M/M/1 queuing system means we have one queue per server. It does not mean that you cannot have multiple servers. A diagram above shows 4 servers with 4 queues. Therefore, each of these servers are computed using M/M/1 queues.
Use the M/M/1 queuing calculator below to experiment to solve queuing problem of a single server. For instance, what happen to the queuing performance if you can improve the service rate by 20%? You can also compare the performance of 4 servers with 4 queues (4*M/M/1) with the performance of 4 servers with 1 queue (M/M/4). Check the queuing calculator for M/M/s here .
M/M/1 Queuing system is also equivalent to M/G/1 queuing system with standard deviation of service rate . The formulation of measurement of effectiveness of M/M/1 queuing system are given below.
Input:
 Arrival rate (number of customers/unit time)
 Service rate (number of customers/unit time)
Output:
 U= Utilization factor = percentage of the time that all servers are busy,
 P _{ 0 } = probability that there are no customers in the system,
 P _{ n } = probability that there are n customers in the system,
 W _{ q } = average time a customer spends in waiting line waiting for service,
 W = average time a customer spends in the system (in waiting line and being served),
 L _{ q } = average number of customer in waiting line for service,
 L = average number of customer in the system (in waiting line and being served),
Example:
Our observations of one cashier in a supermarket have shown that the arrival distribution of customers follows Poisson distribution with arrival rate of customers/5 minutes. The distribution of service time follows Exponential distribution with average service time = 1.079 minutes per customer. Calculate the measurement of effectiveness of the queuing system.
Solution:
This problem is M/M/1 queuing system because we have only single server with Poisson arrival distribution and Exponential service time distribution. The arrival rate is customers/5 minutes = 0.15 customers/minute. Putting the same unit, we have average service time of = 1.079 minutes per customer which is equivalent to customers per minute.
Utilization factor = percentage of the time that all servers are busy,
P _{ 0 } = probability that there are no customers in the system,
Average time a customer spends in waiting line waiting for service, minutes.
Average time a customer spends in the system (in waiting line and being served), minutes.
L _{ q } = average number of customer in waiting line for service, customers.
L = average number of customer in the system (in waiting line and being served), customers.
P _{ n } = probability that there are n customers in the system,
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Do you have queuing problem? Consult your expert for a solution here
These tutorial is copyrighted .
Preferable reference for this tutorial is
Teknomo, Kardi. (2014) Queuing Theory Tutorial
http://people.revoledu.com/kardi/tutorial/Queuing/