Question

Consider a sequential-service system consisting of two servers, \(A\) and \(B\). Arriving customers will enter this system only if server \(A\) is free. If a customer does enter, then he is immediately served by server \(A\). When his service by \(A\) is completed, he then goes to \(B\) if \(B\) is free, or if \(B\) is busy, he leaves the system. Upon completion of service at server \(B\), the customer departs. Assume that the (Poisson) arrival rate is two customers an hour, and that \(A\) and \(B\) serve at respective (exponential) rates of four and two customers an hour. (a) What proportion of customers enter the system? (b) What proportion of entering customers receive service from B? (c) What is the average number of customers in the system? (d) What is the average amount of time that an entering customer spends in the system?

Short answer

(a) 50% of customers enter the system. (b) 100% of entering customers receive service from server B. (c) The average number of customers in the system is 1/3 customers. (d) The average amount of time an entering customer spends in the system is 1/6 hours.

Step-by-step solution

Calculate the Utilization Rates

Firstly, we need to calculate the utilization rates for both servers A and B. The utilization rate is the ratio of the arrival rate to the service rate. Let's denote Utilization rates of server A and B as ρ_A and ρ_B: ρ_A = (arrival rate of clients) / (service rate of server A) ρ_B = (arrival rate of clients who get service in A) / (service rate of server B) Since we are given that the arrival rate is 2 customers per hour, the service rate of server A is 4 customers per hour, and the service rate of server B is 2 customers per hour, we can plug these values into the above formulas.

Calculate the Proportion of Customers Entering the System

To find the proportion of customers entering the system, we can simply find the utilization rate of server A. Based on the calculations done in Step 1, the proportion of customers entering the system (ρ_A) is: ρ_A = 2 / 4 = 1/2 (50% of customers enter the system)

Calculate the Proportion of Entering Customers Receiving Service from B

To calculate the proportion of entering customers receiving service from B (ρ_B), we also need to take into account the successful service rate at server A. Since server A serves at a rate of 4 customers an hour and the arrival rate to the system is 2 customers an hour, the arrival rate to server B is also 2 customers an hour. Hence, ρ_B = 2 / 2 = 1 (100% of entering customers receive service from server B)

Calculate the Average Number of Customers in the System

We can calculate the average number of customers in the system (L) using Little's formula: L = (λ * W_A) + (ρ_B * W_B) Where λ is the arrival rate of customers to the system, W_A and W_B are the average waiting time at servers A and B, respectively. For an M/M/1 queue, the average waiting time in the queue (W) is given by: W = ρ / (service rate * (1 - ρ)) Now, we can calculate W_A and W_B using the above formula and the utilization rates of server A and B found in Steps 1 and 3: W_A = (1/2) / (4 * (1 - 1/2)) = 1/6 hours W_B = (1) / (2 * (1 - 1)) = Infinity (as ρ_B = 1) Since a customer can't wait an infinite amount of time, the average waiting time at server B will be considered as 0 in this case. Now we can plug in the values into the Little's formula for L: L = (2 * 1/6) + (1 * 0) = 1/3 customers

Calculate the Average Time an Entering Customer Spends in the System

The average time an entering customer spends in the system (W_total) can be calculated by adding the average waiting time at servers A and B: W_total = W_A + W_B = 1/6 hours To sum up, (a) The proportion of customers entering the system is 50%. (b) The proportion of entering customers receiving service from B is 100%. (c) The average number of customers in the system is 1/3 customers. (d) The average amount of time that an entering customer spends in the system is 1/6 hours.

Key concepts

Poisson Arrival Rate

The Poisson arrival rate is a key concept in queueing theory, describing how customers or events arrive at a service system over time. It is widely used in various fields such as telecommunications, retail, and transportation because it effectively models random events that occur independently and at a constant average rate. In the context of our sequential-service system exercise, the Poisson arrival rate quantifies the frequency at which customers reach the system, specified as two customers per hour.

This constant rate implies that the time between successive arrivals follows an exponential distribution, which is memoryless. This means the probability of the next arrival does not depend on when the previous arrival occurred. In mathematical terms, if the arrival rate is denoted by \(\lambda\), the probability of seeing \(x\) arrivals in a given time frame \(t\) can be expressed using the Poisson probability formula:
\[ P(X=x) = \frac{e^{-\lambda t}(\lambda t)^{x}}{x!} \]
Understanding the Poisson arrival rate is crucial as it helps in determining service requirements like the number of servers needed and the size of waiting areas.

Exponential Service Rates

Exponential service rates are an assumption of the time it takes for a service episode in a queueing process. In our system with two servers, A and B, the service times are exponentially distributed with rates of four and two customers per hour respectively. This implies that the amount of time one customer is served is random but has a certain average rate at which the service occurs.

The memoryless property of the exponential distribution comes into play here as well; no matter how long server A has been serving a customer, the expected time until the service is completed remains the same. Mathematically, the probability that a service time exceeds a certain time \(t\) is defined as:
\[ P(T > t) = e^{-\mu t} \]
where \(\mu\) represents the service rate. This service rate insight gives us a clue about how our system performance, as measured by customer wait times and server utilization rates, will behave. It is particularly important when analyzing the efficiency of service and predicting the system's capacity to handle the incoming traffic of customers.

Queueing Theory

Queueing theory is a pillar of operations research and it deals with the study of waiting lines or queues. Through this mathematical theory, we can analyze a wide range of systems where congestion and waiting lines form, such as in our textbook example of a sequential-service system with two servers. Queueing theory provides a framework to evaluate system performance by measuring metrics like the average number of customers in the system, customer wait times, and the proportion of time a server is busy versus idle.

Key Metrics and Formulas

In queueing theory, several metrics are commonly used:

• Utilization rate (\(\rho\)): The fraction of time a server is busy.
• Little's Law: Relates the average number of customers in the system (L), the average arrival rate (\(\lambda\)), and the average time a customer spends in the system (W) through the equation \(L = \lambda W\).
• Service rate (\(\mu\)): The rate at which servers can serve customers.

The application of these metrics in formulas allows to gain insights on how efficiently a service system operates and to identify potential improvements. For instance, by knowing the arrival and service rates, we can assess whether a server is overburdened or underutilized and adjust accordingly to optimize the flow and reduce wait times for customers. This theoretical framework is invaluable for designing and managing systems where the flow of entities requires management for efficiency and effectiveness.