Queuing Theory 2014 - Exercises

The importance for this tutorial is to introduce the students into the loss systems, and, to underline the difference between the concepts of call-blocking and time-blocking probabilities, and to understand under which conditions these probabilities are identical. The selected exercises introduce, also, the Erlang tables, which is an important tool for easy calculations for the blocking probabilities. 6.1 Exercise 6.5 A telephone switch has 10 output lines and a large number of incoming lines. Upon arrival a call on the input line is assigned an output line if such line is available – otherwise the call is blocked and lost. The output line remains assigned to the call for its entire duration which is of exponentially distributed length. Assume that 180 calls / hour arrive in Poisson fashion whereas the mean call duration is 110 seconds. 1. Determine the blocking probability. 2. How many calls are rejected per hour? 3. What is the average load per server (in Erlang)? 4. What is the maximum arrival rate at which a blocking probability of (at most) 2% can be guaranteed? Solution: This is an introductory problem for the Markovian Loss Systems. We start with the system mode and the Kendall Notation: • Poisson arrivals with rate λ = 180 calls per hour • Exponential service times with rate μ = 1 E[T ] = 3600 110 • 10 Servers • No buffer The Kendall Notation is: M/M/10/10. We draw the system diagram (Fig. 8) and, based on that, we build the balance equations to compute the state probabilities. A state, S, defines the number of active calls in the system. We define