/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 52 Consider a system where the inte... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 52

Consider a system where the interarrival times have an arbitrary distribution \(F\), and there is a single server whose service distribution is \(G\). Let \(D_{n}\) denote the amount of time the \(n\) th customer spends waiting in queue. Interpret \(S_{n}, T_{n}\) so that $$ D_{n+1}=\left\\{\begin{array}{ll} D_{n}+S_{n}-T_{n}, & \text { if } D_{n}+S_{n}-T_{n} \geqslant 0 \\ 0, & \text { if } D_{n}+S_{n}-T_{n}<0 \end{array}\right. $$

Short Answer

Expert verified
In the given equation for the waiting time of customers in a single server system with interarrival time distribution \(F\) and service time distribution \(G\), the variables \(S_n\) and \(T_n\) can be interpreted as follows: - \(S_n\) represents the service time for the \(n\)th customer. - \(T_n\) represents the interarrival time between the arrival of the \(n\)th customer and the arrival of the \((n+1)\)th customer.

Step by step solution

01

The variable \(D_n\) represents the amount of time the \(n\)th customer spends waiting in the queue. Our goal is to interpret the variables \(S_n\) and \(T_n\) in the given equation. #Step 2: Identify the role of Sn and Tn in the equation#

The equation is: $$ D_{n+1}=\left\{\begin{array}{ll} D_{n}+S_{n}-T_{n}, & \text { if } D_{n}+S_{n}-T_{n} \geqslant 0 \\\ 0, & \text { if } D_{n}+S_{n}-T_{n}<0 \end{array}\right. $$ This is a piecewise function, which means that the equation for \(D_{n+1}\) depends on whether or not the expression \(D_n + S_n - T_n\) is non-negative. #Step 3: Analyzing the two cases#
02

We have two cases: 1. If \(D_n + S_n - T_n \geq 0:\) In this case, the waiting time for the \((n+1)\)th customer is equal to the waiting time of the \(n\)th customer, plus some additional time \(S_n\), minus some other time \(T_n\). Here, the \((n+1)\)th customer's waiting time depends on the \(n\)th customer's waiting time, which means the server is still working on the \(n\)th customer when the \((n+1)\)th customer arrives. Hence, \(S_n\) would represent the service time of the \(n\)th customer, and \(T_n\) would represent the interarrival time between the \(n\)th and \((n+1)\)th customers. 2. If \(D_n + S_n - T_n < 0:\) In this scenario, the \((n+1)\)th customer's waiting time is zero, meaning they have no delay. This tells us that the server finished working on the \(n\)th customer before the \((n+1)\)th customer arrives. In this case, \(S_n\) represents the service time of the \(n\)th customer, and \(T_n\) is the interarrival time between the \(n\)th and \((n+1)\)th customers, similar to the previous case. #Step 4: Interpreting Sn and Tn#

Based on the analysis in step 3, we can interpret \(S_n\) and \(T_n\) as follows: - \(S_n\) is the service time for the \(n\)th customer. - \(T_n\) is the interarrival time between the arrival of the \(n\)th customer and the arrival of the \((n+1)\)th customer.

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Most popular questions from this chapter

A supermarket has two exponential checkout counters, each operating at rate \(\mu\). Arrivals are Poisson at rate \(\lambda\). The counters operate in the following way: (i) One queue feeds both counters. (ii) One counter is operated by a permanent checker and the other by a stock clerk who instantaneously begins checking whenever there are two or more customers in the system. The clerk returns to stocking whenever he completes a service, and there are fewer than two customers in the system. (a) Let \(P_{n}=\) proportion of time there are \(n\) in the system. Set up equations for \(P_{n}\) and solve. (b) At what rate does the number in the system go from 0 to \(1 ?\) from 2 to \(1 ?\) (c) What proportion of time is the stock clerk checking? Hint: Be a little careful when there is one in the system.

In an \(M / G / 1\) queue, (a) what proportion of departures leave behind 0 work? (b) what is the average work in the system as seen by a departure?

The economy alternates between good and bad periods. During good times customers arrive at a certain single-server queueing system in accordance with a Poisson process with rate \(\lambda_{1}\), and during bad times they arrive in accordance with a Poisson process with rate \(\lambda_{2}\). A good time period lasts for an exponentially distributed time with rate \(\alpha_{1}\), and a bad time period lasts for an exponential time with rate \(\alpha_{2}\). An arriving customer will only enter the queueing system if the server is free; an arrival finding the server busy goes away. All service times are exponential with rate \(\mu\). (a) Define states so as to be able to analyze this system. (b) Give a set of linear equations whose solution will yield the long run proportion of time the system is in each state. In terms of the solutions of the equations in part (b), (c) what proportion of time is the system empty? (d) what is the average rate at which customers enter the system?

A facility produces items according to a Poisson process with rate \(\lambda .\) However, it has shelf space for only \(k\) items and so it shuts down production whenever \(k\) items are present. Customers arrive at the facility according to a Poisson process with rate \(\mu .\) Each customer wants one item and will immediately depart either with the item or empty handed if there is no item available. (a) Find the proportion of customers that go away empty handed. (b) Find the average time that an item is on the shelf. (c) Find the average number of items on the shelf. Suppose now that when a customer does not find any available items it joins the "customers' queue" as long as there are no more than \(n-1\) other customers waiting at that time. If there are \(n\) waiting customers then the new arrival departs without an item. (d) Set up the balance equations. (e) In terms of the solution of the balance equations, what is the average number of customers in the system?

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, she 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, she 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 \(n\) times.
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