...
for j =1, 2, ..., P, where:
Example 1: How does the mean waiting time for class 1 customers compare to that of class 2 customers?
Given there are two classes of traffic (P = 2), let class 1 be standard data and class 2 be urgent data.
where:
As we would expect, note that:
Example 2: What are N, Nq, T, and W for each class of customer and for the system as a whole?
Consider a single-server queueing system with a nonpreemptive priority queueing discipline and two classes of customers. The arrivals for each class are Poisson with rates λ1 = 2 arrivals/second, and λ2 = 1 arrival/second, respectively.
Class 1 customers have exponential service times with an average service time of 1/4 seconds, while class 2 customers have deterministic service with a service time of 1/4 seconds. This implies μ1 = 4 services/second and μ2 = 2 services/second.
First note that:
As such, the queueing system is stable.
Class 1 Customers (w/exponential service times)
Class 2 Customers (w/deterministic service times)
So:
and:
and:
Putting it all together
Thus, we can use Little's Law to find the Nq for each class of customer:
and:
Likewise, we can solve for T and N of each class:
To analyze the entire system (comprised of both customer classes), we can solve for W, T, N, and Nq:
