Consider a single-server queueing model in which customers arrive according to a
ID: 3597538 • Letter: C
Question
Consider a single-server queueing model in which customers arrive according
to a nonhomogeneous Poisson process. Upon arriving they either enter service
if the server is free or else they join the queue. Suppose, however, that each
customer will only wait a random amount of time, having distribution F,
in queue before leaving the system. Let G denote the service distribution.
Define variables and events so as to analyze this model, and give the updating
procedures. Supposewe are interested in estimating the average number of lost
customers by time T , where a customer that departs before entering service is
considered lost.
Explanation / Answer
Customers arrive at the café and join the line. The time between any two customer arrivals is variable and uncertain but we can say, for example, on average two customers join the line every minute. The rate at which customers join the line is what we call arrival rate (symbol lambda ). This is the same as workload arrival in chapter 1. In this case, arrival rate lambda is 2 per minute. This also means that the average time between two arrivals is ½ minute; we call this interarrival time.
therefore, variable and uncertain. Let us say, on average, service time for a customer is 15 seconds = ¼ minute. This also means that the one cashier can serve customers at the rate of 1/(1/4) = 4 customers per minute; this is what we call service rate (symbol mu ).
Since there is only one cashier, number of cashiers is 1. We use symbol m for this. In case cafe opens another cashier and if a single line is used to feed both servers, we will say number of servers m is 2. In case there are two different lines in front of two cashiers then we will say that these are two different queues, each with m=1. For now, let us go with our original scenario of single queue with single server.
First output we can get is the utilization of our resource, the cashier. We use the symbol rho for the utilization. This is equal to the ratio of the rate at which work arrives and the capacity of the station. The idea is exactly the same as utilization computation in chapter 1.We know that the rate at which work arrives is arrival rate lambda . One cashier can service the work at the rate of service rate . If there are more than one servers (that is, if number of servers m is more than 1) then total rate at which work can be served, station capacity, will be m multiplied by. Therefore, utilization rho can be calculated as lambda divided by (m multiplied by). . Make sure arrival rate and service rate are expressed in same unit of time; for example, both should be per minute or per hour. In the case of Orin café, Our cashier is busy 50% of the times. For our calculations to work, utilization rho must be less than 1.