Characteristics of queuing model
Arrival rate / Arrival Pattern: The no. of customers arriving per unit time is termed as arrival rate. Customer arrival is random & therefore it is assumed to follow the Poisson distribution.
Service Rate or Service Pattern (μ): The no. of customers serviced per unit time is known as service rate and it is assumed to follow exponential distribution (negative exponential).
Service Rule or Service Order:
It gives information about the queue discipline which means the order by which customers are picked up from the waiting line in order to provide service. E.g.
(1). FIFO or FCFS: First Come first-out or First come first served
(2). LCFS: Last come first served
(3). SIRO: Service in random order
(4). Priority treatment.
System and Calling population:
System: It is the place or facilities where the customers arrive in order to get service and its capacity may be finite or infinite.
Calling population: In entire sample of customer from which few visit the system is termed as calling population.
Representation of queuing model
Queuing model is represented in Kendall & Lee notation whose general form is
Where a: probability distribution for arrival pattern
b: probability distribution for service pattern
c: no of server within the system
d: service rule or service order
e: size or capacity of system
f: size or capacity of calling population or input source.
(i) If λ > μ:
Queue length will keep on increasing and after certain period of time incoming population will not get serviced. In this condition we can’t find the solution as system ultimately fails.
(ii). If λ < μ → System works
λ < μ → prefer
- The ratio of arrival to service rate indicate the % time server is busy it is known as utilization factor or avg. utilization or system utilization or channel efficiency or cleaning ratio.
- It also indicates the probability that a new customer will have to wait.
Probability that the system is idle or Prob. of zero customer in the system.
Probability of having exactly ‘n’ customer in the system
Avg. no. of customers in the system:
In these we include both the customers waiting in the queue along with those getting service.
Avg. no. of customers in the queue
In these we do not include the customer getting serviced
Avg. waiting time of customers in the queue
The mean waiting time in queue is given by
Avg. waiting time of customers in the system
The mean waiting time in system is given by
Little’s Law
For a stable system avg no. of customer in the system or queue is given by
Ws: avg. waiting time of customer in the system
Wq: avg. waiting time of customer in the queue.
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