WHAT IS THE MEAN QUEUE LENGTH OF THE SYSTEM?STATS 320 Assignment 4 Due: 2pm, Thu 4 Jun 2015 1. [12 marks] Visitors arrive at a security checkpoint according to a Poisson process of rate 2 per minute.

WHAT IS THE MEAN QUEUE LENGTH OF THE SYSTEM?STATS 320 Assignment 4 Due: 2pm, Thu 4 Jun 2015 1. [12 marks] Visitors arrive at a security checkpoint according to a Poisson process of rate 2 per minute.

STATS 320 Assignment 4 Due: 2pm, Thu 4 Jun 2015 1. [12 marks] Visitors arrive at a security checkpoint according to a Poisson process of rate 2 per minute. There is only one security guard at service and the individual service times are only known to be random with a mean 25 seconds and a standard deviation 40 seconds. (a) [3 marks] Explain why this can not be an M/M/1 queue. (b) [3 marks] Find the proportion of the time the security guard is busy. (c) [3 marks] Suppose the checkpoint opens at 8am. How long on average should a visitor who arrives in the afternoon wait in the queue before being examined? (d) [3 marks] How long on average does the security guard need to work continuously before having a break? 2. [20 marks] Cars arrive at a police breath alcohol testing station in a Poisson stream with rate 5 per minute. Suppose there are two policemen available for testing the drivers and the time (in minutes) for a policeman to conduct the test has an Erlang distribution Γ(2, 3). To not inconvenience drivers too much, if there are 6 cars in the queue (including being tested), arriving cars are waved through without joining the queue. All inter-arrival times and service times are independent. Let N(t) be the number of cars in the queue at time t (minutes), starting with N(0) = 0. Further, let L(t) = E{N(t)}, i.e., the expected number of cars in the queue at time t. (a) [10 marks] Write an R programme that simulates this queue, and plot a random realization of N(t) for the first 20 minutes. (b) [5 marks] Based on simulation, estimate L(2) and L(5). (c) [5 marks] Based on simulation, estimate L in steady-state. 1 3. [25 marks] In this question, we investigate the effectiveness of ramp metering, using a very simplistic model. It is simplistic because only a road between points A and B is considered. At point A, there is a traffic signal that may be switched on so that one vehicle is allowed to enter the road when the light turns green, or switched off so that vehicles can enter without restriction. During rush hours, vehicles arrive at point A in a Poisson stream with rate λ per minute. Inter-departure times of vehicles at point B follow an exponential distribution with rate µ per minute that depends on N(t), the number of vehicles on the road at time t, as follows: µ(N) = µ0 × N N0 × α(N) where α(N) = 1, if N ≤ N0; exp{β(1 − N/N0)}, otherwise.

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