Question

The economy alternates between good and bad periods. During good times customers arrive at a certain single-server queueing system in accordance with a Poisson process with rate \(\lambda_{1}\), and during bad times they arrive in accordance with a Poisson process with rate \(\lambda_{2} .\) A good time period lasts for an exponentially distributed time with rate \(\alpha_{1}\), and a bad time period lasts for an exponential time with rate \(\alpha_{2}\). An arriving customer will only enter the queueing system if the server is free; an arrival finding the server busy goes away. All service times are exponential with rate \(\mu .\) (a) Define states so as to be able to analyze this system. (b) Give a set of linear equations whose solution will yield the long-run proportion of time the system is in each state. In terms of the solutions of the equations in part (b), (c) what proportion of time is the system empty? (d) what is the average rate at which customers enter the system?

Short answer

The queueing system can be analyzed by defining 4 states: \(S(0,0)\), \(S(1,0)\), \(S(0,1)\), and \(S(1,1)\), representing the number of customers in the system (0 or 1) and the economy condition (good or bad). Linear equations can be set up to find the long-run proportion of time spent in each state: 1. \(P_{00}(\lambda_2 + \alpha_1) = P_{10}\mu + P_{01}\alpha_2\) 2. \(P_{10}(\mu + \alpha_1) = P_{00}\lambda_2\) 3. \(P_{01}(\lambda_1 + \alpha_2) = P_{11}\mu + P_{00}\alpha_1\) 4. \(P_{11}(\mu + \alpha_2) = P_{01}\lambda_1\) 5. \(P_{00} + P_{10} + P_{01} + P_{11} = 1\) The proportion of time the system is empty is given by \(P_E = P_{00} + P_{01}\). The average customer arrival rate can be found as \(\overline{\lambda} = P_{00}\lambda_2 + P_{01}\lambda_1\). Solve the system of linear equations for the long-run probabilities and substitute the values into the corresponding equations to find the proportion of time the system is empty and the average customer arrival rate.

Step-by-step solution

(Define States)

To analyze this queueing system, we need to define the states. Let's denote the state by \(S(n,g)\), where \(n\) represents the number of customers in the system (either 0 or 1, since new customers only join if the server is free), and \(g\) is a binary indicator for the economy condition (1 for good times, and 0 for bad times). This gives us 4 possible states: - State \(S(0,0)\): No customer in the system, and the economy is in a bad state. - State \(S(1,0)\): One customer in the system, and the economy is in a bad state. - State \(S(0,1)\): No customer in the system, and the economy is in a good state. - State \(S(1,1)\): One customer in the system, and the economy is in a good state.

(Set up Linear Equations)

We will now set up linear equations representing the flow of probabilities between these states. Let \(P_{ij}\) denote the long-run proportion of time spent in state \(S(i,j)\). For state \(S(0,0)\): \(P_{00}(\lambda_2 + \alpha_1) = P_{10}\mu + P_{01}\alpha_2\) For state \(S(1,0)\): \(P_{10}(\mu + \alpha_1) = P_{00}\lambda_2\) For state \(S(0,1)\): \(P_{01}(\lambda_1 + \alpha_2) = P_{11}\mu + P_{00}\alpha_1\) For state \(S(1,1)\): \(P_{11}(\mu + \alpha_2) = P_{01}\lambda_1\) Additionally, we have the normalization constraint: \(P_{00} + P_{10} + P_{01} + P_{11} = 1\)

(Proportion of Time System is Empty)

To find the proportion of time the system is empty, we need to consider the probabilities of states \(S(0,0)\) and \(S(0,1)\), which represent the cases when no customers are in the system. Therefore, we'll add the probabilities as follows: \[P_E = P_{00} + P_{01}\]

(Average Customer Arrival Rate)

To find the average customer arrival rate, we need to calculate the weighted average of the arrival rates in good and bad times, based on the proportion of time in each state. That is: \(\overline{\lambda} = \frac{P_{00}\lambda_2 + P_{01}\lambda_1}{P_{00} + P_{01} + P_{10} + P_{11}}\) Since we are given that \(P_{00} + P_{01} + P_{10} + P_{11} = 1\), this simplifies to: \(\overline{\lambda} = P_{00}\lambda_2 + P_{01}\lambda_1\) To calculate the proportion of time system is empty and the average customer arrival rate, we must first solve the system of linear equations for the long-run probabilities (\(P_{00}, P_{01}, P_{10}, P_{11}\)) and then substitute the values into the corresponding equations.