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Chapter 8: Problem 43

In a two-class priority queueing model suppose that a cost of \(C_{i}\) per unit time is incurred for each type \(i\) customer that waits in queue, \(i=1,2\). Show that type 1 customers should be given priority over type 2 (as opposed to the reverse) if $$ \frac{E\left[S_{1}\right]}{C_{1}}<\frac{E\left[S_{2}\right]}{C_{2}} $$

Short Answer

Expert verified
The given inequality states that type 1 customers should be given priority over type 2 customers if: \[ \frac{E\left[S_{1}\right]}{C_{1}} < \frac{E\left[S_{2}\right]}{C_{2}} \] By comparing the expected total waiting costs for both scenarios, we find that when the given condition is true, the expected total waiting cost in Scenario A (type 1 customers given priority) is lower than the expected total waiting cost in Scenario B (type 2 customers given priority): \[ \text{Cost}_A < E\left[S_{2}\right]C_{2} - E\left[S_{2|1}\right]C_{2} \] Thus, we have demonstrated that type 1 customers should be given priority over type 2 customers when the given inequality holds.

Step by step solution

01

Review the given inequality

We are given the condition that type 1 customers should be given priority if: \[ \frac{E\left[S_{1}\right]}{C_{1}} < \frac{E\left[S_{2}\right]}{C_{2}} \] Where \(E[S_i]\) represents the expected waiting time for type \(i\) customers and \(C_i\) represents the cost incurred per unit time for each type \(i\) customer in the queue.
02

Calculate expected total waiting costs

We need to determine which scenario has a lower expected total waiting cost. Let's consider: 1. Scenario A: Type 1 customers are given priority 2. Scenario B: Type 2 customers are given priority We calculate expected total waiting costs for both scenarios. In Scenario A, the expected total waiting cost can be calculated as: \[ \text{Cost}_A = E\left[S_{1}\right]C_{1} + E\left[S_{2|1}\right]C_{2} \] And in Scenario B: \[ \text{Cost}_B = E\left[S_{1|2}\right]C_{1} + E\left[S_{2}\right]C_{2} \] where \(E[S_{2|1}]\) denotes the expected waiting time for type 2 customers in Scenario A (when type 1 customers are given priority) and \(E[S_{1|2}]\) denotes the expected waiting time for type 1 customers in Scenario B (when type 2 customers are given priority).
03

Compare expected total waiting costs

We now look for the scenario with the lower expected total waiting cost, where we substitute the given inequality. Let's rewrite the inequality to make it in terms of cost: \[ E\left[S_{1}\right]C_{1} < E\left[S_{2}\right]C_{2} \] And rearrange the inequality: \[ E\left[S_{1}\right]C_{1} - E\left[S_{2}\right]C_{2} < 0 \] Now, we can subtract \(E\left[S_{2|1}\right]C_{2}\) from both sides: \[ E\left[S_{1}\right]C_{1} + E\left[S_{2|1}\right]C_{2} - E\left[S_{2}\right]C_{2} < -E\left[S_{2|1}\right]C_{2} \] Recall the definition of Scenario A's cost, which is: \[ \text{Cost}_A = E\left[S_{1}\right]C_{1} + E\left[S_{2|1}\right]C_{2} \] Using this, our inequality can be re-written as: \[ \text{Cost}_A - E\left[S_{2}\right]C_{2} < -E\left[S_{2|1}\right]C_{2} \] Finally, add \(E\left[S_{2}\right]C_{2}\) to both sides: \[ \text{Cost}_A < E\left[S_{2}\right]C_{2} - E\left[S_{2|1}\right]C_{2} \] Recall that for Scenario B, the cost is: \[ \text{Cost}_B = E\left[S_{1|2}\right]C_{1} + E\left[S_{2}\right]C_{2} \] The inequality shows that when the given condition is true, the expected total waiting cost in Scenario A (type 1 customers given priority) is lower than the expected total waiting cost in Scenario B (type 2 customers given priority). Thus, it is shown that type 1 customers should be given priority over type 2 when the given inequality holds.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cost Analysis
When analyzing a priority queueing model, cost analysis plays a critical role in determining how priorities should be assigned among different customer types. Imagine every customer patiently waiting in a queue like customers do at a supermarket, and think about the cost associated for how long they wait. The exercise specifically looks into how to decide which type of customer should get priority based on minimizing costs.
  • Each customer type (like type 1 or type 2) incurs a specific cost per unit time, denoted as \(C_i\).
  • The main idea is to evaluate the overall cost associated with different priority schemes, like giving priority to type 1 over type 2.
  • The exercise simplifies the decision process by analyzing the inequality involving expected waiting times and costs for each customer type.
This helps in making informed decisions on queue management, ensuring that costs remain as low as possible by prioritizing the right set of customers in line.
Expected Waiting Time
In queueing models, expected waiting time is an essential metric. Simply put, it refers to the average amount of time a customer of a particular type spends waiting in the queue before receiving service. Each customer type has a different expected waiting time, which helps in assessing how long, on average, each customer will wait based on their priority level.
  • Represented as \(E[S_i]\), where \(i\) refers to the customer type, it provides a way to quantify waiting time in mathematical terms.
  • The expected waiting time directly impacts the cost per unit time due to the inequality \(\frac{E[S_1]}{C_1} < \frac{E[S_2]}{C_2}\), guiding priority assignment.
  • By understanding expected waiting times, businesses can design efficient queue systems, ensuring customer satisfaction and operational cost-effectiveness.
Thus, expected waiting time is a key component in balancing queue efficiency and service costs.
Queueing Model
A queueing model is a mathematical representation of a real-life queue system. It helps in determining how best to service customers who are lined up or waiting. Queueing models are like roadmaps for service operations; they provide a structured way to examine and optimize the flow of customers.
  • These models consider different customer types and their assigned priority levels.
  • They evaluate how service policies impact waiting times and costs within the system.
  • The models aim to minimize downtime and expense by articulating scenarios where certain customers should take precedence.
In this exercise, the queueing model is crucial for setting priorities by comparing different service strategies and aligning them with cost considerations. This involves not only tracking who waits for how long but ensuring that it's done in a way that aligns with minimizing financial impacts.
Customer Type Priority
Establishing customer type priority is foundational in queueing systems. Different customer types may bring different values and costs, necessitating a strategy to determine which group should receive priority in receiving service. Imagine having a 鈥榝ast lane鈥� in a service line catering especially to high-priority customers!
  • Customer type priority ensures that those incurring more significant costs or demanding quicker service are handled first for efficiency and satisfaction.
  • The priority decision is rested upon the cost-benefit analysis, where the expected reduction in cost drives the priority choice.
  • Mathematically, this may be reflected in systems using inequations like \(\frac{E[S_1]}{C_1} < \frac{E[S_2]}{C_2}\), meaning prioritize type 1 customers over type 2 if the inequality holds true.
Customer type priority helps businesses manage queues better, reducing overall wait times and meeting customer expectations effectively without increasing operational costs disproportionately.

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Most popular questions from this chapter

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?

A group of \(m\) customers frequents a single-server station in the following manner. When a customer arrives, he or she either enters service if the server is free or joins the queue otherwise. Upon completing service the customer departs the system, but then returns after an exponential time with rate \(\theta\). All service times are exponentially distributed with rate \(\mu\). (a) Define states and set up the balance equations. In terms of the solution of the balance equations, find (b) the average rate at which customers enter the station. (c) the average time that a customer spends in the station per visit.

Machines in a factory break down at an exponential rate of six per hour. There is a single repairman who fixes machines at an exponential rate of eight per hour. The cost incurred in lost production when machines are out of service is \(\$ 10\) per hour per machine. What is the average cost rate incurred due to failed machines?

Customers arrive at a two-server system at a Poisson rate \(\lambda\). An arrival finding the system empty is equally likely to enter service with either server. An arrival finding one customer in the system will enter service with the idle server. An arrival finding two others in the system will wait in line for the first free server. An arrival finding three in the system will not enter. All service times are exponential with rate \(\mu\), and once a customer is served (by either server), he departs the system. (a) Define the states. (b) Find the long-run probabilities. (c) Suppose a customer arrives and finds two others in the system. What is the expected times he spends in the system? (d) What proportion of customers enter the system? (e) What is the average time an entering customer spends in the system?

Customers arrive at a two-server station in accordance with a Poisson process with a rate of two per hour. Arrivals finding server 1 free begin service with that server. Arrivals finding server 1 busy and server 2 free begin service with server 2. Arrivals finding both servers busy are lost. When a customer is served by server 1 , she then either enters service with server 2 if 2 is free or departs the system if 2 is busy. A customer completing service at server 2 departs the system. The service times at server 1 and server 2 are exponential random variables with respective rates of four and six per hour. (a) What fraction of customers do not enter the system? (b) What is the average amount of time that an entering customer spends in the system? (c) What fraction of entering customers receives service from server \(1 ?\)
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