Question: Consider a first-in first-out single-server queueing system in which customers arrive according to a Poisson process with rate 2 per time unit. Each customer

Consider a first-in first-out single-server queueing system in which customers arrive according to a Poisson process with rate 2 per time unit. Each customer requires exponential service time with mean 1/ time units. The system experiences occasional failures, and when the system fails all the customers in the system are dropped (including the customer currently under service). The time distribution between failures is exponentially distributed with mean 1/y time units. Assume that all the random variables are independent. a) Show that the system can be modeled using a Markov chain and draw a state-diagram. b) Write the balance equations for the state-diagram. c) Assume =u-y-1. Find the stationary distribution of the number of customers in the queue. Hint: Guess a solution of the form 7(i)= (1-a)a', where (i) represents the stationary probability of finding i customers in the system and a is a variable which value must be determined. d) Assume 2--y-1. Find the fraction of customers that successfully complete service.

Step by Step Solution

★★★★★

3.51 Rating (151 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Operating System Questions!

Northwood Company manufactures a basketball selling for $25 per unit in a small plant heavily relying on direct labor workers. Thus, variable expenses are high, totaling $15.00 per ball, of which 60%...

Consider a queue to which customers arrive according to a Poisson process with rate per hour. Suppose that the queue has two servers. Each customer who arrives at the queue counts the length r of...

Consider a queueing system with two types of customers. Type 1 customers arrive according to a Poisson process with a mean rate of 5 per hour. Type 2 customers also arrive according to a Poisson...

The truss ABC shown in the figure supports a horizontal load P1 = 300 lb and a vertical load P2 = 900 lb. Both bars have cross-sectional area A = 2.4 in.2 and are made of steel with E = 30 Ã...

Two charges are placed as shown in Fig. 17-35 with q1 = 1.5μC and Q2 = 3.3μC. Find the potential difference between points A and B. 91 0.10 m 13 0.10 m

Develop a list of strategic issues that must be addressed in performing facilities planning for a. an airport b. a community college c. a bank d. a grocery store chain e. a soft drink bottler and...

6.30 Loading Grain A grain loader can be set to discharge grain in amounts that are normally distributed, with mean kg and standard deviation equal to 700 kg. If a company wishes to use the loader to...

Case-Based Marketing Selling on Amazon.com A case-based reasoning system is a type of expert system. It attempts to match the facts on hand to a database of prior cases. When a case-based reasoning...

At each calendar year-end, Mazie Supply Co. uses the percent of accounts receivable method to estimate bad debts. On December 31, 2013, it has outstanding accounts receivable of $74,000, and it...

Morrow Enterprises Inc. manufactures bathroom fixtures. The stockholders' equity accounts of Morrow Enterprises Inc., with balances on January 1, 20Y5, are as follows: Common Stock, $20 stated value...

suppose T(suppose T(n) and g(n) are positive functions thatsatisfy T(n) =k and some constant C>0prove that there is a constant C??>0 for which T(n)

After researching the different forms of business organization, Natalie Koebel decides to operate "Cookie Creations" as a proprietorship. She then starts the process of getting the business running....

P. 6-1 The financial statements of an actual capital projects fund leave it to the report reader to draw inferences on key transactions. The accompanying statements of the parks, recreations, and...

An infinite solid circular cylinder is initially at a uniform temperature of 150 C. At time t = 0 the temperature around the entire boundary is suddenly reduced to 0C, and maintained thereafter....

6. Germs Co. revalued its building with historical cost of P40,000,000 and accumulated depreciation of P10,000,000 to a fair value of P48,000,000. Income tax rate is 30%. Requirements: a. Compute for...

In a report, 2-3 pages single-spaced, describe the purpose of the audit software and describe its capabilities.Provide at least two examples of how companies have used the software as it relates to...

The number of employees in an organization increased due to expansion and introduction of new products. The top management decided that the organization will be divided into different groups. Few...

Give the products of the following reaction, where T is tritium: dioldehydrase Ad- CH CH3C-COH CoIII) coenzyme B12

Consider a three-echelon inventory system that fits the model for a serial multiechelon system presented in Sec. 18.5, where the model parameters for this particular system are given below....

The demand D for a product in a week is a random variable taking on the values of 1, 0, 1 with probabilities 1/8, 5/8, and C/8, respectively. A demand of 1 implies that an item is returned. (a) Find...

This case is a continuation of Case 4.3, which involved the Springfield School Board assigning students from six residential areas to the citys three remaining middle schools. After solving a linear...

3. A colleague of yours claims that animals have emotions just as humans do. How would you test that?

3. Ever notice how people see the world differently? How do people come to have their views of the world? Their perceptions? Take any of the figures in the section on optical illusions and conduct...

4. What kinds of stereotypes do you have of people whose mannerisms are different than yours? How would you feel if you spoke with someone who didnt look you in the eye when talking with you?