345. M/M/c
- M: Memoryliss interarrival times
- M: Memoryless service times
- c: number of servers
Input parameters:
-
: number of parallel servers in the system
-
: average service rate of μ (mu) per server
- Exponential distribution: customers per unit of time
-
: average rate of arrivals per unit of time
- Poisson process: number of arrivals in a given time period
- Exponential distribution: time between successive arrivals
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Exponential Distribution: Rate parameter λ = 2 (same as above) |
Poisson Distribution: Rate parameter , so |
System Stability
-
:
- Arrival rate greater than service rate
- Queue will grow infinitely
-
:
- Without variability: no queue or wait time
- With variability: may have wait time
- :
Performance Metrics
- Utilization ()
Where:
- : arrival rate (mean number of arrivals per unit of time)
- : service rate (mean number of services completed per server per unit of time)
- Probability of Zero Customers in the System (Idle Probability) ()
Probability that no customers are in the system
- Probability that All Servers are Busy (Blocking Probability or Queueing Probability) ()
Probability that all servers are busy and a customer will have to wait in the queue
- Average Number of Customers in the System ()
Average number of customers in the system (both in the queue and being served)
- Average Number of Customers in the Queue ()
Average number of customers waiting in the queue (not being served)
- Average Time a Customer Spends in the System ()
Average time a customer spends in the system (both waiting and being served) (Little’s Law)
- Average Time a Customer Spends in the Queue ()
Average waiting time a customer spends in the queue before being served