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Chapter 8: Problem 18

Suppose in Exercise 17 we want to find out the proportion of time there is a type 1 customer with server \(2 .\) In terms of the long-run probabilities given in Exercise 17 , what is (a) the rate at which a type 1 customer enters service with server \(2 ?\) (b) the rate at which a type 2 customer enters service with server \(2 ?\) (c) the fraction of server 2 's customers that are type \(1 ?\) (d) the proportion of time that a type 1 customer is with server \(2 ?\)

Short Answer

Expert verified
(a) The rate at which a type 1 customer enters service with server 2 is given by: Rate = \(P_{1,2} \cdot 位_1\) (b) The rate at which a type 2 customer enters service with server 2 is given by: Rate = \(P_{2,2} \cdot 位_2\) (c) The fraction of server 2's customers that are type 1 is given by: Fraction = \(\frac{P_{1,2} \cdot 位_1}{(P_{1,2} \cdot 位_1) + (P_{2,2} \cdot 位_2)}\) (d) The proportion of time that a type 1 customer is with server 2 is given by: Proportion of time = \(\frac{P_{1,2} \cdot 位_1}{(P_{1,2} \cdot 位_1) + (P_{2,2} \cdot 位_2)}\)

Step by step solution

01

Understand the given information from Exercise 17

From Exercise 17, we have the long-run probabilities as follows: \(P_{1,2}\) - The probability that there is a type 1 customer with server 2. \(P_{2,2}\) - The probability that there is a type 2 customer with server 2. \(位_1\) - The arrival rate of the type 1 customer. \(位_2\) - The arrival rate of the type 2 customer. Now let's find the rates and proportions.
02

Calculate the rate at which a type 1 customer enters service with server 2 (part a)

To calculate this, we will multiply the probability of a type 1 customer with server 2 by the arrival rate of type 1 customers. Rate = \(P_{1,2} \cdot 位_1\)
03

Calculate the rate at which a type 2 customer enters service with server 2 (part b)

Similarly, we will multiply the probability of a type 2 customer with server 2 by the arrival rate of type 2 customers. Rate = \(P_{2,2} \cdot 位_2\)
04

Find the fraction of server 2's customers that are type 1 (part c)

To find this, we will add the rate of entry of type 1 customers with server 2, and the rate of entry of type 2 customers with server 2. We will then divide the rate of entry of type 1 customers with server 2 by the total rate of entry. Fraction = \(\frac{P_{1,2} \cdot 位_1}{(P_{1,2} \cdot 位_1) + (P_{2,2} \cdot 位_2)}\)
05

Calculate the proportion of time that a type 1 customer is with server 2 (part d)

This can be found by dividing the rate at which a type 1 customer enters service with server 2 by the sum of the rates at which both types of customers enter server 2. Proportion of time = \(\frac{P_{1,2} \cdot 位_1}{(P_{1,2} \cdot 位_1) + (P_{2,2} \cdot 位_2)}\) This proportion of time is the same as the fraction calculated in Step 4. It represents the proportion of time that a type 1 customer is with server 2 in the long run.

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Key Concepts

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

Long-Run Probabilities
When studying probability models in the context of queues or services, long-run probabilities are a pivotal concept. These probabilities represent the steady state of a system when it has been running for a sufficiently long time, so much so that the initial conditions have no bearing on the current state.

For example, in the given exercise, the long-run probabilities such as \(P_{1,2}\) for a type 1 customer being with server 2, help determine the behavior and utilization of resources over an extended period. By understanding these long-run figures, organizations can optimize their service processes and plan resources accordingly.
Arrival Rate
In service and queueing models, the arrival rate is a fundamental parameter that measures the frequency at which customers arrive at the service system. It is often denoted by \(\lambda\) and could differ for different types of customers or services.

For instance, \(\lambda_1\) could represent the arrival rate of type 1 customers, and this figure helps in calculating various performance metrics of the service system, such as the load on a server or the waiting time for customers. Understanding this parameter is essential for designing a system that can effectively handle the customers and their needs.
Type 1 Customer
Differentiating customers into types is a method used to add complexity and realism to a model. A type 1 customer in a model could have different characteristics such as arrival rates, service requirements, or profitability. For instance, a type 1 customer could be a regular customer whose behavior patterns and requirements are well known to the service provider.

Models that differentiate between customer types allow for more refined analysis of the service systems, enabling businesses to tailor specific strategies for customer satisfaction and resource management.
Type 2 Customer
Similarly, a type 2 customer represents another distinct group with its own arrival rate and service profile. They may differ from type 1 customers in terms of frequency of visits, the complexity of service needed, or their value to the business.

Recognizing and planning for different types of customers within a service model is crucial because it impacts the flow and efficiency of the service process. Businesses can allocate resources effectively and improve service delivery by understanding the characteristics of each customer type.
Server Model
A server model, within the context of queueing theory and probability models, refers to the specifics of how a customer is serviced in a system. It includes aspects like the number of servers, the service rate, the system's capacity, and the discipline of the service (e.g., first-come, first-served).

In our exercise, server 2 is such a model component where both type 1 and type 2 customers are served. Determining metrics like the proportion of time a server spends with a certain type of customer or the rate at which each type of customer is served lends valuable insights for improving service efficiency and customer satisfaction in the long run.

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Most popular questions from this chapter

The manager of a market can hire either Mary or Alice. Mary, who gives service at an exponential rate of 20 customers per hour, can be hired at a rate of \(\$ 3\) per hour. Alice, who gives service at an exponential rate of 30 customers per hour, can be hired at a rate of \(\$ C\) per hour. The manager estimates that, on the average, each customer's time is worth \(\$ 1\) per hour and should be accounted for in the model. If customers arrive at a Poisson rate of 10 per hour, then (a) what is the average cost per hour if Mary is hired? if Alice is hired? (b) find \(C\) if the average cost per hour is the same for Mary and Alice.

Consider a single-server exponential system in which ordinary customers arrive at a rate \(\lambda\) and have service rate \(\mu .\) In addition, there is a special customer who has a service rate \(\mu_{1}\). Whenever this special customer arrives, it goes directly into service (if anyone else is in service, then this person is bumped back into queue). When the special customer is not being serviced, the customer spends an exponential amount of time (with mean \(1 / \theta\) ) out of the system. (a) What is the average arrival rate of the special customer? (b) Define an appropriate state space and set up balance equations. (c) Find the probability that an ordinary customer is bumped \(n\) time.

Compare the M/G/1 system for first-come, first-served queue discipline with one of last-come, first-served (for instance, in which units for service are taken from the top of a stack). Would you think that the queue size, waiting time, and busy-period distribution differ? What about their means? What if the queue discipline was always to choose at random among those waiting? Intuitively which discipline would result in the smallest variance in the waiting time distribution?

Consider a \(M / G / 1\) system with \(\lambda E[S]<1\). (a) Suppose that service is about to begin at a moment when there are \(n\) customers in the system. (i) Argue that the additional time until there are only \(n-1\) customers in the system has the same distribution as a busy period. (ii) What is the expected additional time until the system is empty? (b) Suppose that the work in the system at some moment is \(A\). We are interested in the expected additional time until the system is empty- call it \(E[T] .\) Let \(N\) denote the number of arrivals during the first \(A\) units of time. (i) Compute \(E[T \mid N]\). (ii) Compute \(E[T]\).

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?
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