Question

In a queue with unlimited waiting space, arrivals are Poisson (parameter \(\lambda\) ) and service times are exponentially distributed (parameter \(\mu\) ). However, the server waits until \(K\) people are present before beginning service on the first customer; thereafter, he services one at a time until all \(K\) units, and all subsequent arrivals, are serviced. The server is then "idle" until \(K\) new arrivals have occurred. (a) Define an appropriate state space, draw the transition diagram, and set up the balance equations. (b) In terms of the limiting probabilities, what is the average time a customer spends in queue? (c) What conditions on \(\lambda\) and \(\mu\) are necessary?

Short answer

To analyze the given queueing system, we first defined the state space representing the number of customers in the system. Then, we set up the balance equations for the idle and active server conditions to find the limiting probabilities of each state. We used Little's theorem to compute the average time a customer spends in the queue (W_q) based on the limiting probabilities. Finally, we determined that the condition for stability of the system is λ < μ, meaning the service rate should be greater than the arrival rate.

Step-by-step solution

Define State Space and Transition Diagram

First, we need to define the state space for this queueing system. The states represent the number of customers in the system. It can be represented as {0,1,2,3,...}. A state 'n' means there are n customers in the system. The transition diagram shows the transitions between states. Each arrow between states represents a transition, which can be either a customer arrival with rate λ or a customer service with rate μ. The server will start service only after K customers are present. We cannot draw the transition diagram here, but you can visualize it as a series of states on a line, where each state has a transition to the next state. For example: 0 -> 1 -> 2 -> ... -> K -> K+1 -> ... Now, we set up the balance equations.

Set up the Balance Equations

The balance equations represent the steady-state condition of the system, ensuring that the flow in and out of each state is balanced. We will represent the limiting probability of being in state n as π_n. The balance equations will depend on whether the server is active (serving customers) or idle (waiting for K customers). For n = 0 to K-1 (server is idle, waiting for K customers): π_n * λ = π_(n+1) * μ For n >= K (server is active, serving customers): π_n * λ + π_(n-1) * μ = π_(n+1) * μ + π_n * μ Now, solve these equations for the limiting probabilities π_n.

Average Time Spent in Queue

We need to compute the average time a customer spends in the queue. We can use Little's theorem, which states that the average number of customers in the queue (L_q) is equal to the arrival rate (λ) multiplied by the average time spent in queue (W_q). Mathematically, this is given as: L_q = λ * W_q The average number of customers in the queue, L_q, can be calculated using the limiting probabilities: L_q = \( \sum_{n=1}^{\infty} (n-1) \cdot \pi_n \) From Little's theorem, we then compute the average time a customer spends in the queue (W_q): W_q = L_q / λ

Conditions on λ and μ

To ensure the stability of the queueing system, we need to ensure the following condition is satisfied: λ < μ This ensures that the service rate is greater than the arrival rate, which prevents the system from becoming overwhelmed with customers and ensures that the queue reaches a steady state. In summary, we defined the state space and transition diagram, set up the balance equations, calculated the average time a customer spends in the queue using limiting probabilities, and determined the conditions on λ and μ to ensure the stability of the system.