Machine Repair (M/M/s/N/N) Queuing System

Queuing Tutorial

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M/M/s/N/N Queuing System

In this model of queuing system, the arrival distribution of customers follows Poisson distribution and the distribution for service time follows Exponential distribution with s number of parallel servers. The number of population and the queuing capacity is limited to N. This situation often happens in queuing for machine repair system where the number of population is equal to the number of machine = N.

Use the M/M/s/N/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 and M/M/s/N queuing system.

The performance for M/M/s/N/N 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 = limited population = \( N \)
  • Maximum queue size = \( N - s \)

Output:

  • \( U \) = operative Utilization = \( U=\sum_{i=0}^{s-1} \frac{iP_{i}}{s} + \sum_{j=s}^{N} P_{j} \)
  • \( P_{0} \) = probability that there are no customers in the system, \( P_{0} =\left ( \sum_{i=0}^{s} \binom{N}{i} \rho^{i} + \sum_{j=s+1}^{N} \binom{N}{j} \frac{j!\rho^{j}}{s!s^{j-s}} \right )^{-1} \)
  • \( P_{n} \) = probability that there are n customers in the system,\( P_{n}=\begin{cases} \binom{N}{n} \rho^{n} P_{0} & \text{ if } n\leqslant s \\ \binom{N}{n} \frac{n! \rho^{n}}{s!s^{n-s}} P_{0} & \text{ if } s \leqslant n \leqslant N \\ \end{cases} \)
  • \( W_{q} \) = average time a customer spends in waiting line waiting for service, \( W_{q}=\frac{L_{q}}{\lambda\left ( N-L \right )} \)
  • \( W \) = average time a customer spends in the system (in waiting line and being served), \( W=\frac{L}{\lambda\left ( N-L \right )} \)
  • \( L_{q} \) = average number of customer in waiting line for service, \( L_{q}=\begin{cases} \sum_{i=s+1}^{N}\left ( i-s \right ) P_{i} & \text{ if } s>1 \\ N-\left ( 1+\frac{1}{\rho} \right )\left ( 1-P_{0} \right ) & \text{ if } s=1 \end{cases} \)
  • \( L \) = average number of customer in the system (in waiting line and being served), \( L=\begin{cases} L_{q}+s-\sum_{i=0}^{s}\left ( s-i \right ) P_{i} & \text{ if } s>1 \\ N-\frac{1-P_{0}}{\rho} & \text{ if } s=1 \end{cases} \)
  • Machine availability \( 1 - \frac{L}{N} \)


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These tutorial is copyrighted .

Preferable reference for this tutorial is

Teknomo, Kardi. (2014) Queuing Theory Tutorial
http://people.revoledu.com/kardi/tutorial/Queuing/