Question

There are two types of customers. Type 1 and 2 customers arrive in accordance with independent Poisson processes with respective rate \(\lambda_{1}\) and \(\lambda_{2}\). There are two servers. A type 1 arrival will enter service with server 1 if that server is free; if server 1 is busy and server 2 is free, then the type 1 arrival will enter service with server 2\. If both servers are busy, then the type 1 arrival will go away. A type 2 customer can only be served by server \(2 ;\) if server 2 is free when a type 2 customer arrives, then the customer enters service with that server. If server 2 is busy when a type 2 arrives, then that customer goes away. Once a customer is served by either server, he departs the system. Service times at server \(i\) are exponential with rate \(\mu_{i}\), \(i=1,2\) Suppose we want to find the average number of customers in the system. (a) Define states. (b) Give the balance equations. Do not attempt to solve them. In terms of the long-run probabilities, what is (c) the average number of customers in the system? (d) the average time a customer spends in the system?

Short answer

To summarize, we defined the following states for our queuing system: state 0 when both servers are empty, state 1 when server 1 is busy and server 2 is empty, state 2 when server 1 is empty and server 2 is busy, and state 3 when both servers are busy. Using these states, we set up the balance equations and found the long-run probabilities. The average number of customers in the system is given by \(L = (0)P_0 + (1)P_1 + (1)P_2 + (2)P_3\), and the average time a customer spends in the system is given by \(W = \frac{L}{\lambda_1 + \lambda_2}\).

Step-by-step solution

1. Define States

To define the states, consider the following possible states of the system: - State 0: Both servers are empty. - State 1: Server 1 is busy, Server 2 is empty. - State 2: Server 1 is empty, Server 2 is busy. - State 3: Both servers are busy. The system is in state 0 if no customers are present; state 1 when a type 1 customer is being served by server 1; state 2 when a type 2 customer is served by server 2 or type 1 customer is served by server 2; and state 3 when server 1 serves a type 1 customer and server 2 serves a type 1 or type 2 customer.

2. Balance Equations

To derive the balance equations, we consider the inflow and outflow from the states. Let \(P_i\) be the long-run probability that the system is in state i. - For State 0: \[\lambda_1(P_1 - P_0) + \lambda_2(P_2 - P_0) = 0\] - For State 1: \[\lambda_1(P_0 - P_1) + (\lambda_1 + \lambda_2)(P_3 - P_1) + \mu_1(P_3 - P_1) = 0\] - For State 2: \[\lambda_1(P_3 - P_2) + \lambda_2(P_0 - P_2) + \mu_2(P_3 - P_2) = 0\] - For State 3: \[\lambda_1(P_1 - P_3) + \mu_1(P_1 - P_3) + \lambda_2(P_2 - P_3) + \mu_2(P_2-P_3) = 0\] Since these probabilities are long-run probabilities, we also have the normalization condition: \[P_0 + P_1 + P_2 + P_3 = 1\]

3. Average Number of Customers in System

The average number of customers in the system can be found by multiplying the number of customers in each state by the long-run probability of that state and totaling the result: \[L = (0)P_0 + (1)P_1 + (1)P_2 + (2)P_3\]

4. Average Time a Customer Spends in the System

We can use Little's theorem to find the average time a customer spends in the system, which states that \(L = \lambda W\), where \(L\) is the average number of customers in the system, \(\lambda\) is the total arrival rate, and \(W\) is the average time a customer spends in the system. In this case, the total arrival rate, \(\lambda = \lambda_1 + \lambda_2\). To find the average time a customer spends in the system, we rearrange the equation and solve for \(W\): \[W = \frac{L}{\lambda} = \frac{L}{\lambda_1 + \lambda_2}\]

Key concepts

Balance Equations

In queueing theory, balance equations are pivotal in determining the long-term behavior of a system. They are used to model the flow of customers or packets in and out of various states within the system. Suppose we compare the rate at which customers leave each state with the rate at which they enter. The system is in equilibrium if these rates balance out, meaning the inflow and outflow are equal.

The exercise presents a system with two different types of customers and two servers, each with varied service rules. Formulating the balance equations involves a step by step assessment of the inflow and outflow probabilities for each state of the system. We use the rates at which the customers arrive and the servers process them, denoted by \(\lambda_i\) for arrival rates and \(\mu_i\) for service rates. The established balance equations do not require immediate solving but set the stage for understanding the system's dynamics and can later be used to calculate various performance metrics such as the average number of customers in the system and the average time they spend in the system.

Exponential Service Times

Understanding the Exponential Distribution

Exponential service times are a common assumption in the study of queueing systems, which implies the memoryless property – the probability of a service completion in the next instant is independent of how long the service has already been in process.

This characteristic is especially relevant in the given problem where each server processes customers one at a time and the service time is exponentially distributed with rates \(\mu_1\) and \(\mu_2\). This assumption not only simplifies the analysis by using these rates directly in the balance equations but also allows the application of Markovian properties for solving the queueing system.

Queueing Theory

Queueing theory is a mathematical study of waiting lines, or queues. At its core, it employs mathematical models to analyze various aspects of queues such as waiting times, queue lengths, and the efficiency of the queueing process.

The given exercise exemplifies a basic queue model where customers arrive according to a Poisson process. This is defined by random events occurring independently and at a constant average rate, which are common criteria in many real-world queueing scenarios. Queueing theory often leverages these stochastic processes to predict system behavior over time and is an essential tool in designing and managing systems where resources are shared, such as in customer service centers, computer networks, and manufacturing.

Little's Theorem

Little's theorem is a fundamental theorem in queueing theory that provides a critical link between the average number of customers in a system (L), the average arrival rate of customers to the system (\(\lambda\)), and the average time a customer spends in the system (W).

The theorem, stated simply as L = \(\lambda W\), applies to a wide range of queueing systems, under the assumption of steady-state conditions. In our exercise, the average arrival rate and the average number of customers are known, and thus Little's theorem can directly provide the average time a customer spends in the system. It's a powerful result that is widely used due to its general applicability and simplicity.