3. Jobs arrive in a computer queue in the manner of a Poisson process with intensity λ. The centralprocessor handles them one by one in the order of their arrival, and each has an exponentially distributedruntime with parameter , the runtimes of different jobs being independent of each other and of thearrival process. Let X (t) be the number of jobs in the system (either running or waiting) at time t,where X (0) = 0. Explain why X is a Markov chain, and write down its generator. Show that astationary distribution exists if and only if λ < μ, and find it in this case.

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Answered: 3. Jobs arrive in a computer queue in…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 2EQ: 2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed...

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2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.7. How many bacteria of each strain can coexist in the test tube and consume all of the food? Table 2.7 Bacteria Strain I Bacteria Strain II Bacteria Strain III Food A 1 2 0 Food B 2 1 3 Food C 1 1 1
Consider a stable single-station queue. Suppose it costs $c to hold a customer in the system for one unit of time. Show that the long-run holding cost rate is given by cL, where L is the mean number of customers in steady state.
Two teams A and B play a soccer match. The number of goals scored by Team A is modeled by a Poisson process N₁ (t) with rate X₁ = 0.02 goals per minute, and the number of goals scored by Team B is modeled by a Poisson process N₂ (t) with rate A₂ = 0.03 goals per minute. The two processes are assumed to be independent. Let N(t) be the total number of goals in the game up to and including time t. The game lasts for 90 minutes. a. Find the probability that no goals are scored, i.e., the game ends with a 0-0 draw. b. Find the probability that at least two goals are scored in the game. c. Find the probability of the final score being Team A: 1, Team B: 2 d. Find the probability that they draw.
Suppose that a patient receives a daily dose of 50mg/L of a certain drug such that 42% of it is eliminated from the body each day. If on a certain Monday, the concentration of the drug measured in their body (shortly after the daily dose) is 55 mg/L, write down the Discrete-Time Dynamical System describing the dynamics of the concentration xt of the drug in the body (in mg/L, t days after that Monday). Then write down the general solution to this DTDS.
Suppose that each year, 80% of residents living in state A stay in state A, while 20% of them move to state B. Also, 70% of the residents living in state B stay in state B, while 30% of them move to state A. Further, assume at t = 0 years, there are 2 million residents living in state A and 5 million residents living in state B, and let xn and yYn represent the populations (in millions) of the states A and B, respectively, at time t = n years. Given that (:) - (: )() Xn a b Yn d 5 find the value of the number h.
EXAMPLE 6.56 If λ = 2 per hour and u = 4 per hour in a 2-capacity single- server queueing system, find the effective arrival rate.
The value y (in 1982–1984 dollars) of each dollar paid by consumers in each of the years from 1994 through 2008 in a country is represented by the ordered pairs. (1994, 0.675) (1995, 0.654) (1996, 0.643) (1997, 0.618) (1998, 0.611) (1999, 0.599) (2000, 0.585) (2001, 0.562) (2002, 0.559) (2003, 0.539) (2004, 0.529) (2005, 0.513) (2006, 0.493) (2007, 0.482) (2008, 0.462) (b) Use the regression feature of the spreadsheet software program to find a linear model for the data. (Let trepresent time. Round your numerical values to four decimal places.) y = (c) Use the model to predict the value (in 1982–1984 dollars) of 1 dollar paid by consumers in 2010 and in 2011. (Round your answers to two decimal places.)
An electrical system consists of 2 components (C1 and C2) functioning independently and are set in a serial layout. The failure time of each component follows an exponential distribution with rate 2 in every year. Thus the system will operate if both components function, and itwill fail if any one component fails.(a) Obtain the failure density and distribution of the system(b) Obtain the survival function/distribution of the system.(c) Based on your answer in (a) and (b), calculatei) the probability that the system fails in less than 8 months.ii) the expected time to failure of the system.
Suppose that each year, 80% of residents living in state A stay in state A, while 20% of them move to state B. Also, 70% of the residents living in state B stay in state B, while 30% of them move to state A. Further, assume at t = 0 years, there are 2 million residents living in state A and 5 million residents living in state B, and let æn and yn represent the populations (in millions) of the states A and B, respectively, at time t = n years. Given that Xn a 2 Yn d find the value of the number c.
The reproduction number R of an epidemic spreading process taking place on a random network with degree distribution P(k) is given by R=A(k(k-1)) (k) where k indicates the degree of the nodes and the average (...) indicates the average over the degree distribution, P(k). Therefore R is the product between the infectivity of the virus, due to its biological fitness and the branching ratio of the network, depending on the degree distribution of the network and given by (k(k-1))/(k). According to the value of R the epidemic can be in different regimes: If R > 1 the epidemics is in the supercritical regime: the epidemics spreads on a finite fraction of the population, resulting in a pandemics. If R < 1 the epidemics is in the subcritical regime: the epidemics affects a infinitesimal fraction of the population and can be considered suppressed. • If R = 1 the epidemics is in the critical regime: this is the regime that separates the previous two regimes. Consider an epidemics with…
An average of 125 packets of information per minutearrive at an internet router. It takes an average of .002second to process a packet of information. The router isdesigned to have a limited buffer to store waiting messages.Any message that arrives when the buffer is full is lost tothe system. Assuming that interarrival and service times are exponentially distributed, how big a buffer size is needed toensure that at most 1 in a million messages is lost?
In this problem, assume Poisson arrivals and exponential service times.Customers of Plaka Record rental outlet arrive at the store to rent CDs with an arrivalrate of 1.25 customers per minute. The store has only 1 counter with a service rate of 2 customers per minute. Suppose that Plaka Record rental outlet adds another counter to serve thecustomers simultaneously.1. The average number of customers in the waiting line2. The average number of customers in the system3. The average time a customer spends in the waiting line. The average time a customer spends in the system

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