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M/M/s/N Queuing System
The previous M/M/s queuing system assume that you have unlimited space for the customers to queue. That is of course unrealistic. M/M/s/N queuing system is probably more realistic model because it set the capacity of the system at N. However, this model must be used with care because the meaning of capacity in here is a real cut off of customers, not the actual waiting room capacity. Under this model, if a customer arrives after the maximum queue size is reached, then that customer must be rejected by the system as a loss customer. If you have a waiting room capacity but you can still make the customers to queue outside on the road, probably M/M/s queuing system is still a good model.
The arrival distribution of customers follows Poisson distribution and the distribution for service time follows Exponential distribution with s number of parallel servers and N queuing capacity.
Use the M/M/s/N queuing calculator below to experiment and to solve queuing problem of multiple parallel servers with queuing capacity N. Compare it with M/M/s with M/M/s/N/N queuing system.
The performance for our queuing system are given by the formulas below.
Input:
- Arrival rate (number of customers/unit time) \( \lambda \)
- Service rate (number of customers/unit time) \( \mu \)
- Number of servers \( s \)
- Capacity of the system, \( N \)
- Maximum queue size = \( N - s \)
Output:
- \( U \) = Utilization factor = percentage of the time that all servers are busy, \( U=\frac{L-L_{q}}{s} \)
- \( P_{0} \) = probability that there are no customers in the system, \( P_{0} =\left ( 1+ \sum_{i=1}^{s} \frac{\rho^{i}}{i!} + \frac{\rho^{s}}{s!} \sum_{j=s+1}^{N} \left ( \frac{\rho}{s}\right ) ^{j-s} \right )^{-1} \)
- \( P_{n} \) = probability that there are n customers in the system, \( P_{n}=\begin{cases} \frac{\rho^{n}}{n!} P_{0} & \text{ if } 1 \leq n \leq s \\ \frac{\rho^{n}}{s!s^{n-s}} P_{0} & \text{ if } s+1 \leq n \leq N \\ 0 & \text{ if } n>N \end{cases} \)
- \( W_{q} \) = average time a customer spends in waiting line waiting for service, \( W_{q}=\frac{L_{q}}{\lambda\left ( 1-P_{N} \right )} \)
- \( W \) = average time a customer spends in the system (in waiting line and being served), \( W=W_{q}+\frac{1}{\mu} \)
- \( L_{q} \) = average number of customer in waiting line for service, \( L_{q}=\sum_{n=s}^{N} \left ( n-s \right ) P_{n} \)
- \( L \) = average number of customer in the system (in waiting line and being served), \( L=\sum_{n=0}^{N} n P_{n} \)
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Do you have queuing problem? Ask 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/