Question

Consider a closed queueing network consisting of two customers moving among two servers, and suppose that after each service completion the customer is equally likely to go to either server-that is, \(P_{1,2}=P_{2,1}=\frac{1}{2}\). Let \(\mu_{i}\) denote the exponential service rate at server \(i, i=1,2\) (a) Determine the average number of customers at each server. (b) Determine the service completion rate for each server.

Short answer

The short answer based on the given step-by-step solution is: The average number of customers at Server 1: \(L_1 = 4 \times P(\text{state 1}) \) The average number of customers at Server 2: \(L_2 = 3 - L_1\) The service completion rate for Server 1: \(\lambda_1 = L_1 \times \mu_1\) The service completion rate for Server 2: \(\lambda_2 = L_2 \times \mu_2\)

Step-by-step solution

Identify the system states

Our system (queueing network) consists of two servers and two customers. The customers can be in any of the following configurations: 1. Both customers at server 1. 2. One customer at server 1 and one customer at server 2. 3. Both customers at server 2. These three configurations represent the possible states of our system. STEP 2: Create the state transition diagram

Create the state transition diagram

Now that we've identified the states, we can create a state transition diagram. This will help us visualize the different possible transitions between states. Each state will be represented by a node, and directed edges will represent the possible transitions accompanied by the probabilities of such transitions. In our case, the transitions among states 1 and 3 will always involve passing through state 2. There are only two transitions that directly involve state 1 and state 3, so we can denote the transitions as shown: 1 --(1/2,μ₁)--> 2 --(1/2,μ₂)--> 3 2 --(1/2,μ₂)--> 1 --(1/2,μ₁)--> 2 Now that we have our state transition diagram, we can proceed to find the average number of customers at each server and the service completion rates. STEP 3: Calculate the average number of customers at each server:

Calculate the average number of customers at each server

To calculate the average number of customers at each server, we consider the different states of our system and calculate the proportion of time that each state occurs using the balance equations. Let L₁ and L₂ denote the averages number of customers at Server 1 and Server 2. Then, L₁ = P(state 1) × 2 + P(state 2) × 1 L₂ = P(state 2) × 1 + P(state 3) × 2 Since the system is a closed queueing network with an equal probability of customers going to either server, then P(state 1) = P(state 3). So, we obtain the following balance equations: L₁ = 2 × P(state 1) + P(state 2) L₂ = P(state 2) + 2 × P(state 3) Since the three probabilities P(state 1), P(state 2) and P(state 3) must add up to 1, we can solve the equations above to get: L₁ = 4 × P(state 1) L₂ = 3 - L₁ This will give us the average number of customers at each server. STEP 4: Calculate the service completion rates

Calculate the service completion rates

Service completion rate for each server is related to the service rate μ, which is given as μ₁ for server 1 and μ₂ for server 2. Since the system has an exponential service rate and is in steady-state, the service completion rate equals the service arrival rate. Thus, Service completion rate for Server 1: λ₁ = L₁ × μ₁ Service completion rate for Server 2: λ₂ = L₂ × μ₂ This will give us the service completion rates for each server.

Key concepts

Closed Queueing Network

Imagine a miniature ecosystem where a fixed number of customers visit various service stations, never leaving the system and constantly moving from one service station to another. This is the essence of a closed queueing network. In this closed system, the total number of customers remains constant; whenever a customer leaves a service station, they are reassigned to another, maintaining the network's integrity.

This type of network is particularly useful in scenarios where resources are limited and need to be allocated efficiently, such as in computer networks, manufacturing systems, or even amusement park designs where cars or seats are continuously cycled through different rides or stages of a process.

Exponential Service Rate

The exponential service rate is a critical concept in queueing theory that describes the time between services completed at a station. It follows an exponential probability distribution, which is commonly associated with the memoryless property.

It means that the probability of a service being completed in the next instant is independent of how long the current service has already taken. This simplifies many calculations in queueing theory and the analysis of systems. In the context of our exercise, the service rate at each server is exponential, signified with the symbol \(\mu_i\) for the ith server.

State Transition Diagram

To visualise the labyrinth of potential states and transitions in a queueing system, we use the state transition diagram. This diagram serves as a map, marking each state as a node and the pathways between these states as edges, with each path assigned a probability and rate at which transitions occur.

By mapping out the landscape of the network through this state transition diagram, we gain a visual and mathematical tool to explore and determine how often each state occurs, which is essential for uncovering metrics like the average number of customers at each server or the service completion rates.

Average Number of Customers in a Queue

Students often grapple with the question of how to ascertain the average number of customers in a queue. This number is pivotal, as it reflects the load on the service station and can indicate the level of congestion or efficiency.

To compute this average, we assess the likelihood and duration of the system being in various states. Each state corresponds to a different number of customers at the server, and by finding a weighted sum of these numbers with respect to the probabilities of the states, we arrive at the average number we cross paths with in each queue.

Service Completion Rate

Moving on from the number of customers is the service completion rate. This tells us how rapidly the servers are dealing with their lines, which is pivotal for understanding the system's flow and capacity management.

To calculate this, we look at the service rates and the average number of customers at each server. Since the service times are exponentially distributed, the completion rate is simply the product of the server's service rate and the average number of customers at that server. This can be used to gauge productivity and identify bottlenecks within the network.

Balance Equations

Imagine a balanced scale, representing the harmony that must exist in a queueing system. The balance equations achieve just that for a closed network by ensuring that the flow into each state matches the flow out.

These equations keep track of the probabilities of each state, guaranteeing that the sum of all probabilities equals 1, since the system must be in one of the states at all times. Balancing these equations is akin to solving a puzzle: correctly placing each piece to create a cohesive, functioning whole.

Steady-State Probabilities

The final piece of our theoretical framework is the steady-state probabilities. These probabilities are the bedrock of queueing theory, representing the long-term behavior of the system.

They tell us the likelihood that the system will be found in any given state after it has been operating for a significant amount of time, having reached a state of equilibrium. These probabilities are crucial for calculating many performance measures of the system, such as the average number of customers or service rates, which remain stable over time in a steady-state scenario.