Question

Consider a set of \(n\) machines and a single repair facility to service these machines. Suppose that when machine \(i, i=1, \ldots, n\), fails it requires an exponentially distributed amount of work with rate \(\mu_{i}\) to repair it. The repair facility divides its efforts equally among all failed machines in the sense that whenever there are \(k\) failed machines each one receives work at a rate of \(1 / k\) per unit time. If there are a total of \(r\) working machines, including machine \(i\), then \(i\) fails at an instantaneous rate \(\lambda_{i} / r\) (a) Define an appropriate state space so as to be able to analyze the preceding system as a continuous-time Markov chain. (b) Give the instantaneous transition rates (that is, give the \(\left.q_{i j}\right)\). (c) Write the time reversibility equations. (d) Find the limiting probabilities and show that the process is time reversible.

Short answer

The continuous-time Markov chain analysis of the given system with \(n\) machines and a repair facility involves defining a state space, finding the instantaneous transition rates, time reversibility equations, and limiting probabilities. The state space \(S = \{0, 1, 2, ..., n\}\) represents the number of failed machines. Instantaneous transition rates \(q_{i, i+1}\) and \(q_{i, i-1}\) can be found by calculating the rate at which a working machine fails and the rate at which a broken machine is repaired, respectively. Time reversibility equations are given by \(\pi_i q_{i, i+1} = \pi_{i+1} q_{i+1, i}\). The limiting probabilities can be found recursively using the normalization condition and time reversibility equations, and the time reversibility of the process can be verified using these probabilities.

Step-by-step solution

(a) Define state space for the system.

Define the state space of the system, such that each state represents the number of failed machines in the system. So, let \(S = \{0, 1, 2, ..., n\}\) be the state space, where \(n\) is the total number of machines. Each state \(s_i\) with \(i \in S\) represents a system with \(i\) machines failed.

(b) Instantaneous transition rates between states.

To determine the instantaneous transition rates, for each state \(s_i\), we need to find the rates of two possible transitions: 1) From state \(s_i\) to state \(s_{i+1}\), which occurs when a working machine fails, and 2) From state \(s_i\) to state \(s_{i-1}\), which occurs when a broken machine is repaired. Let \(q_{i, i+1}\) be the instantaneous transition rate from state \(i\) to state \(i+1\), and \(q_{i, i-1}\) be the instantaneous transition rate from state \(i\) to state \(i-1\). 1) The rate at which a working machine fails is given by \(\lambda_i / r\), where \(r\) is the number of working machines, and \(\lambda_i\) is the failure rate of machine \(i\). Since \(r = n-i\), the failure rate for each working machine should be summed up, i.e., \( q_{i, i+1} = \sum_{j=1}^{n-i} \frac{\lambda_j}{n-i} \) 2) The rate at which a broken machine is repaired is given by \(\mu_i / k\), where \(k\) is the number of broken machines. The repair facility divides its efforts equally among all broken machines, so each receives work at a rate of \(1/k\) per unit time. Thus, \( q_{i, i-1} = i\cdot \frac{\mu_i}{i} = \mu_i \)

(c) Time reversibility equations.

In a continuous-time Markov chain, the process is time-reversible if the product of the limiting probabilities and the transition rates remains constant, i.e., \( \pi_i q_{i, i+1} = \pi_{i+1} q_{i+1, i} \) Substituting the instantaneous transition rates from part (b), \( \pi_i \sum_{j=1}^{n-i} \frac{\lambda_j}{n-i} = \pi_{i+1} \mu_{i+1} \)

(d) Find limiting probabilities and verify time reversibility.

To find the limiting probabilities, we can use the previous equation and the normalization condition - the sum of all probabilities should equal 1: (i) \( \sum_{i=0}^n \pi_i = 1 \) (ii) \( \pi_i \sum_{j=1}^{n-i} \frac{\lambda_j}{n-i} = \pi_{i+1} \mu_{i+1} \) We can recursively compute the probabilities by rearranging equation (ii): \( \pi_{i+1} = \frac{\pi_i \sum_{j=1}^{n-i} \frac{\lambda_j}{n-i}}{\mu_{i+1}} \) Once we obtain all the limiting probabilities, we can use them to verify the time reversibility of the process using the equation obtained in part (c). If the equation is true for all states, then the process is time-reversible.

Key concepts

State Space

When it comes to continuous-time Markov chains, understanding the state space is crucial. It's the foundation upon which we analyze the system's behavior. In our exercise involving a set of machines and a repair facility, the state space is ingeniously defined to represent each possible number of failed machines. Specifically, for a system with n machines, the state space would be S = {0, 1, 2, ..., n}, with each number symbolizing a situation where that many machines have failed. This detailed state space allows us to thoroughly examine the chain's dynamics as machines fail and get repaired.

By meticulously categorizing each state in this manner, the student can visualize and understand the different scenarios the system can be in, aiding both intuition and mathematical analysis.

Instantaneous Transition Rates

Digging deeper into the heart of the problem, we encounter instantaneous transition rates. This is where the action happens — it's the math that brings the chain to life. To put it simply, these rates express how quickly we're likely to move from one state to another in an infinitesimal time frame. For a machine that fails, this rate depends on the number of operating machines, while the repair rate hinges on how many are out of order.

Breaking the transition down:

• When a machine fails, the rate is the sum of failure probabilities for all the working machines.
• For a repair, it's the individual repair rate, because the repair facility works on all failed machines equally.

These rates are crucial because they tell us the 'speed' of state changes and are the building blocks for predicting the system's future.

Time Reversibility

Next, we have time reversibility, a beautiful property that can arise in certain Markov chains. It's a bit like a movie that makes as much sense backwards as forwards. To check if our system is time-reversible, we essentially verify whether the process behaves the same whether we're advancing or rewinding time. The math for this involves combining the limiting probabilities — which show the chain's long-term behavior — with the instantaneous transition rates.

A

Key Equation

is \( \pi_i q_{i, i+1} = \pi_{i+1} q_{i+1, i} \). This must hold true for all states in a reversible system. What it means in practice is that the flow of transitions in one direction is perfectly balanced by the flow in the reverse direction. If these conditions hold, we have not only a fascinating mathematical result but also powerful analytical tools at our disposal.

Limiting Probabilities

Finally, the most anticipated actors in our Markov drama are the limiting probabilities. Just like the ending of a story, these probabilities reveal the eventual state of our system. Intuitively, they are where the system 'settles down' after a long period. These probabilities are found by solving a set of balance equations derived from the aforementioned transition rates.

What's critical is that they add up to one — we're sure our system has to be in one of the states. By calculating these probabilities step by step, we can foresee the long-term behavior of our repair facility and the machines it serves. As an added bonus, if our process is proven time-reversible, it means these probabilities provide an extra layer of symmetry to our system's dynamics.