M/M/c

358. M/M/c

Input parameters:

$𝑐$ : number of parallel servers in the system
$𝜇$ : average service rate of μ (mu) per server
- Exponential distribution: $𝜇$ customers per unit of time
$𝜆$ : average rate of arrivals per unit of time
- Poisson process: number of arrivals in a given time period
- Exponential distribution: time between successive arrivals

Exponential Distribution:
Inter-arrival Times

Rate parameter λ = 2 (same as above)

Poisson Distribution:
Number of Arrivals in Time Period $𝑡 = 2$

Rate parameter $𝜆 = 2$ , so $𝜆 𝑡 = 4$

System Stability

$𝜆 > 𝑐 𝜇$ :
- Arrival rate greater than service rate
- Queue will grow infinitely
$𝜆 < 𝑐 𝜇$ :
- Without variability: no queue or wait time
- With variability: may have wait time
$𝜆 = 𝑐 𝜇$ :

Performance Metrics

𝜌 = \frac{𝜆}{𝑐 \cdot 𝜇}

Where:

$𝜆$ : arrival rate (mean number of arrivals per unit of time)
$𝜇$ : service rate (mean number of services completed per server per unit of time)

Probability that no customers are in the system

𝑃_{0} = {(\sum_{𝑛 = 0}^{𝑐 - 1} \frac{{(\frac{𝜆}{𝜇})}^{2}}{𝑛!} + \frac{{(\frac{𝜆}{𝜇})}^{𝑐}}{𝑐! (1 - 𝑝)})}^{- 1}

Probability that All Servers are Busy (Blocking Probability or Queueing Probability) ( $𝑃_{𝑤}$ )

Probability that all $𝑐$ servers are busy and a customer will have to wait in the queue

𝑃_{𝑤} = \frac{\frac{{(\frac{𝜆}{𝜇})}^{𝑐}}{𝑐!}}{\sum_{𝑛 = 0}^{𝑐 - 1} \frac{\frac{{(\frac{𝜆}{𝜇})}^{𝑛}}{𝑐!}}{𝑛!} + \frac{{(\frac{𝜆}{𝜇})}^{𝑐}}{𝑐! (1 - 𝑝)}}

Average number of customers in the system (both in the queue and being served)

𝐿 = \frac{𝜆}{𝜇} + \frac{𝑃_{𝑤} \cdot (\frac{𝜆}{𝜇})}{𝑐 (1 - 𝑝)}

Average number of customers waiting in the queue (not being served)

𝐿_{𝑞} = 𝑃_{𝑤} \cdot (\frac{𝜌 \cdot (\frac{𝜆}{𝜇})}{1 - 𝜌})

Average time a customer spends in the system (both waiting and being served) (Little’s Law)

𝑊 = \frac{𝐿}{𝜆}

Average waiting time a customer spends in the queue before being served

𝑊_{𝑞} = \frac{𝐿_{𝑞}}{𝜆} = 𝑃_{𝑤} \cdot \frac{1}{𝑐 𝜇 (1 - 𝑝)}