Question

Customers arrive at a single-server station in accordance with a Poisson process with rate \(\lambda\). All arrivals that find the server free immediately enter service. All service times are exponentially distributed with rate \(\mu\). An arrival that finds the server busy will leave the system and roam around "in orbit" for an exponential time with rate \(\theta\) at which time it will then return. If the server is busy when an orbiting customer returns, then that customer returns to orbit for another exponential time with rate \(\theta\) before returning again. An arrival that finds the server busy and \(N\) other customers in orbit will depart and not return. That is, \(N\) is the maximum number of customers in orbit. (a) Define states. (b) Give the balance equations. In terms of the solution of the balance equations, find (c) the proportion of all customers that are eventually served; (d) the average time that a served customer spends waiting in orbit.

Short answer

We define the state of the system as a tuple (i, j), where 'i' represents the server status and 'j' represents the number of customers in orbit. Using steady-state probabilities, we can write the balance equations for a system with given parameters. To find the proportion of served customers, S, and average time a served customer spends waiting in orbit, W, we can use the following formulas: \(S = \frac{\mu}{\lambda} \sum_{j=0}^{N} P(1,j) \) \(W = \frac{ \sum_{j=1}^{N} jP(1,j)}{S\theta} \)

Step-by-step solution

Define states

We can define the state of the system as a tuple (i, j), where 'i' is the status of the server (0 if the server is free and 1 if the server is busy) and 'j' is the number of customers in orbit (from 0 to N).

Balance equations

Let \(P(i, j)\) represent the steady-state probability of being in state (i, j). Using the steady-state probabilities, we can write the balance equations for a system with 1 server, N customers in orbit, and λ, μ, and θ as given parameters. For the case when the server is free (i=0) and there are no customers in orbit (j=0): \( \lambda P(0,0) = \mu P(1,0) \) For the case when the server is busy (i=1) and there are no customers in orbit (j=0): \( (\lambda + \mu)P(1,0) = \lambda P(0,0) + \theta P(1,1) \) For the case when the server is busy (i=1) and there are j customers in orbit (1 ≤ j ≤ N-1): \( (\lambda + \theta + \mu)P(1,j) = \theta P(1,j+1) + \lambda P(0,j) + \theta P(1,j-1) \) For the last customer in orbit (j=N): \( (\lambda + \theta)P(1,N) = \lambda P(0,N) + \theta P(1,N-1) \)

Proportion of all customers that are eventually served

To find the proportion of all customers that are eventually served, we denote by S the proportion served. This can be found using the balance equations, where the arrival rate multiplied by served proportion equals the service rate multiplied by the server busy state. \(S = \frac{\mu}{\lambda} \sum_{j=0}^{N} P(1,j) \)

Average time a served customer spends waiting in orbit

Let W be the average time a served customer spends waiting in orbit. Using Little's Law, which relates the mean time spent waiting in orbit, the mean number of customers in orbit, and the arrival rate in the orbit, we can find the average time a served customer spends waiting in orbit: \(W = \frac{ \sum_{j=1}^{N} jP(1,j)}{S\theta} \)

Key concepts

Stochastic models

Understanding the complexity of real-world processes often requires sophisticated mathematical approaches like stochastic models. These models are equipped to handle uncertainty and randomness as core aspects of the phenomena they describe. They employ the use of probabilities to forecast the outcomes of a system that is under the influence of random factors. In our exercise, the customers arriving at a service station and the uncertain time they spend in orbit before potentially returning to the server are perfect examples of a scenario that stochastic modeling can clarify. This provides insight into the functioning of the system over time, allowing for predictions and informed decision-making.

Balance equations

The heart of understanding stochastic models often lies in what are known as balance equations. These equations represent equilibrium conditions where the amount of something entering a state is equal to the amount leaving it. In the context of our exercise, balance equations are formulated by considering the rates of customers entering and leaving each state of the system. The steady-state probabilities, which describe the long-term behavior of the system, can be found by solving these balance equations. The significance of achieving balance implies that the system has reached a point where the distribution of probabilities over its states remains constant over time.

Exponential distribution

An integral part of many stochastic models is the exponential distribution, which is widely used in scenarios with a memoryless property – meaning the future is independent of the past. In our textbook example, service times and the time customers spend in orbit are assumed to follow exponential distributions characterized by the rates \( \mu \) and \( \theta \) respectively. The key feature of the exponential distribution is its constant hazard rate, which is particularly useful when modeling the time until an event, such as the time until the next customer arrives or the time until a customer in orbit returns.

Steady-state probability

The notion of steady-state probability is central to understanding complex systems that evolve over time. In a steady-state, the system's properties are unchanging because the processes of entering and leaving states are balanced. In the context of the queuing system, steady-state probabilities tell us the likelihood of finding the system in a particular state (e.g., the server being busy and a certain number of customers in orbit) after a long period. These probabilities are essential for deriving other performance measures of the system, such as the proportion of customers who will eventually be served or the average time customers spend in specific parts of the system.

Little's Law

When analyzing queueing systems, an indispensable tool is Little's Law, a formula that provides a relationship between the average number of items in a queuing system, the average time an item spends in the system, and the average rate at which items arrive. In our exercise, Little's Law is used to calculate the average amount of time a served customer spends waiting in orbit, which involves understanding the flow of customers through the system and how long they tend to stay. The application of Little’s Law offers a simple yet powerful way to quantify the experience of customers within a queueing network, encapsulating complex dynamic interactions in a neat, empirically verifiable equation.