Question

Customers arrive at a two-server station in accordance with a Poisson process having rate \(\lambda\). Upon arriving, they join a single queue. Whenever a server completes a service, the person first in line enters service. The service times of server \(i\) are exponential with rate \(\mu_{i}, i=1,2\), where \(\mu_{1}+\mu_{2}>\lambda .\) An arrival finding both servers free is equally likely to go to either one. Define an appropriate continuoustime Markov chain for this model, show it is time reversible, and find the limiting probabilities.

Short answer

The continuous-time Markov chain for the given model can be defined by the total number of customers in the system, and its transition rates can be obtained based on the arrival and service rates. The steady-state probability distribution can be derived from the balance equations, which is given by: π_i = ((μ₁ + μ₂) / λ)ⁱ * π for i ∈ {0, 1, 2, ...} To normalize the probabilities, we use the condition Σπ_i = 1, and find π: π = 1 / Σ ((μ₁ + μ₂) / λ)ⁱ The system is time reversible as the detailed balance condition holds for all state pairs. The limiting probabilities are the same as the steady-state probabilities.

Step-by-step solution

1. Define the Markov Chain

We can define the state of the system as the total number of customers in the queuing system, which includes the customers being served and those waiting in the queue. Let X(t) be the continuous-time Markov chain representing the number of customers in the system at a time t. Since customers arrive following a Poisson process with rate λ, the arrival process is memoryless, and the departure process is also memoryless due to the exponential service times of servers. Thus, X(t) is a continuous-time Markov chain.

2. Identify the Transition Rates

Let q_ij be the transition rate from state i to state j. For our model, we have the following transition rates: 1. q₀₁ = λ, as the system transitions from state 0 to state 1 when a customer arrives. 2. q₁₀ = μ₁ + μ₂, as the system transitions from state 1 to state 0 when either server completes a service. 3. For i ≥ 2, q_i(i-1) = μ₁ + μ₂, as the system transitions from state i to state i-1 when either server completes a service. 4. For i ≥ 1, q_i(i+1) = λ, as the system transitions from state i to state i+1 when a customer arrives.

3. Balance Equations and Steady-State Distribution

To find the balance equations for our model, we look at the rate of probability coming in and going out of each state. Let π_i be the steady-state probability of the system being in state i. By setting up the balance equations for states i = 0, 1, 2, ..., we have the following system of linear equations: 1. π = λπ₁ / (μ₁ + μ₂) 2. π₁ = μ₁π / λ + μ₂π / λ 3. For i ≥ 2, π_i = (μ₁ + μ₂)π_i-1 / λ Solving this system of linear equations gives us the steady-state probability distribution: For i ∈ {0, 1, 2, ...}, π_i = ((μ₁ + μ₂) / λ)ⁱ * π We can normalize the probabilities with the condition Σπ_i = 1 to find π: 1 = Σ ((μ₁ + μ₂) / λ)ⁱ * π π = 1 / Σ ((μ₁ + μ₂) / λ)ⁱ

4. Time Reversibility and Limiting Probabilities

To show that the system is time reversible, we need to show that the detailed balance condition holds for all state pairs: π_i * q_ij = π_j * q_ji We can easily show this for our model by checking each state pair using the derived steady-state distribution π_i and the transition rates q_ij from Step 2. Since the detailed balance condition holds, the continuous-time Markov chain is time reversible. The limiting probabilities are the same as the steady-state probabilities, given by: π_i = ((μ₁ + μ₂) / λ)ⁱ * π for i ∈ {0, 1, 2, ...}

Key concepts

Poisson Process

The Poisson process is a stochastic model that describes events occurring randomly over time, such as customers arriving at a service station. It is characterized by a constant mean rate \( \lambda \), which is the average number of events happening per unit of time. In the context of our queuing system, this rate is the arrival rate of customers. A key property of the Poisson process is that it is memoryless. This means that the probability of an event occurring in a given interval of time is independent of the past. For example, no matter how many customers have arrived before, the probability of a new customer arriving within the next minute remains the same.
A fascinating aspect of the Poisson process is that the inter-arrival times, or the time between successive events (customer arrivals, in this case), follow an exponential distribution. This linkage between the Poisson process and exponential distribution forms the foundation of many queuing theory models and is critical for analyzing the behavior of queues.

Exponential Service Times

Service times that are exponential with rates \( \mu_{i} \) imply that the time each server takes to serve a customer is random but follows a particular kind of probability distribution. The exponential distribution is, like the Poisson process, memoryless. This signifies that the probability of service being completed in the next moment is the same regardless of how long the service has already been in progress.
The rates \( \mu_{1} \) and \( \mu_{2} \) correspond to the servicing abilities of the two servers in our system. The higher the rate, the quicker the server can process customers. The sum of these rates \( \mu_{1} + \mu_{2} \) surpassing the arrival rate \( \lambda \) is a necessary condition to ensure that the system does not spiral into an infinitely long queue. It is this exponential characteristic of service times that allows the Markov chain to adequately represent the queuing system and for the steady-state probabilities to be determined.

Steady-State Probability

Steady-state probability is a critical concept that refers to the long-run likelihood of the system being in a particular state. In our two-server queue example, \( \pi_{i} \) represents the steady-state probability of having \( i \) customers in the system. This is what the queuing system 'settles down' to after a long period.
It's important to note that we are able to identify a steady-state probability due to the system's balance. The flow of probability into each state from its neighboring states is equal to the flow out of the state. By setting up balance equations, we ensure that the incoming and outgoing rates are equal for all states, allowing us to find the \( \pi_{i} \) values.
Once the steady-state probabilities are determined by solving the balance equations, one can compute the entire distribution by finding \( \pi_{0} \) and normalizing the probabilities so that their sum equals one. These probabilities give us insight into the typical operating characteristics of the system. For instance, they can tell us what fraction of time the system will be completely empty or how often we can expect both servers to be busy.

Time Reversibility

Time reversibility of a Markov chain is an intriguing property indicating that the process behaves the same way backward in time as it does forward. In mathematical terms, a continuous-time Markov chain is reversible if the detailed balance condition is fulfilled for all pairs of states, which means that for every transition from state \( i \) to \( j \) with a steady-state probability \( \pi_{i} \) and rate \( q_{ij} \) there's a corresponding transition from \( j \) to \( i \) with a steady-state probability \( \pi_{j} \) and rate \( q_{ji} \) such that \( \pi_{i} \cdot q_{ij} = \pi_{j} \cdot q_{ji} \).
The significance of time reversibility lies in its implications for system analysis and performance. It implies that the long-run behavior of the Markov chain is insensitive to the direction of time, making the analysis often much simpler and providing valuable symmetries. This property is particularly beneficial when determining the limiting probabilities of the system, as it provides shortcuts to compute the long-term behavior of the Markov chain.