![Consider a M/M/1/2 queue with λ-4 packets/sec and μ-10 packets/sec. <a> Formulate the Markov Chain for the queue. [1 pt] <b> Write the balance equations for the queue. [2 pt] <c> Using the equations in <b>, find the values of P(0), P(l)& P(2). [3 pt] <d> Compute the loss probability. [1 pt] <e> Compute the average population in the system N. [2 pt] Φ Compute the average total delay in the system .[2 pt] Note: You should derive all equations and should not use any ready-made expressions except the Littles Theorem. Also show all your work.](http://img.homeworklib.com/questions/afd69d50-51d8-11eb-bd0b-6b0c54bf3921.png?x-oss-process=image/resize,w_560)
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Consider a M/M/1/2 queue with λ-4 packets/sec and μ-10 packets/sec. <a> Formulate the Markov Chain for...
1. Consider the M/M/1 queue where the arrive rate is λ and the service rate is...
1. Consider the M/M/1 queue where the arrive rate is λ and the service rate is μ (a) Give conditions on and such that the stationary distribution exists (b) For the rest of this problem, assume the stationary distribution exists. Calculate the stationary distribution (c) What is the expected number of individuals in the system at a given time? (d) When a new customer arrives into the queue, how long would they be expected to wait until the leave the...
4. Consider an irreducible Markov chain with finite state space S = {0, 1, , (a)...
4. Consider an irreducible Markov chain with finite state space S = {0, 1, , (a) Starting at state i, what is the probability that it will ever visit state j? (i,j arbi trary (b) Suppose that Xjj iyi for al i. Let ai P(visit N before 0 start at i). Show uations that the r, satisfy, and show that Xi . H2nt: Derive a system of linear eq that xi- solves these equations
2. (10 points) Consider a continuous-time Markov chain with the transition rate matrix -4 2 2 Q 3...
2. (10 points) Consider a continuous-time Markov chain with the transition rate matrix -4 2 2 Q 34 1 5 0 -5 (a) What is the expected amount of time spent in each state? (b) What is the transition probability matrix of the embedded discrete-time Markov chain? (c) Is this continuous-time Markov chain irreducible? (d) Compute the stationary distribution for the continuous-time Markov chain and the em- bedded discrete-time Markov chain and compare the two
2. (10 points) Consider a...
Consider the M/M/1/GD/∞/∞ queuing system where λ and μ are the arrival and server rate, respectively....
Consider the M/M/1/GD/∞/∞ queuing system where λ and μ are the arrival and server rate, respectively. Suppose customers arrive according to a rate given by λ = 12 customers per hour and that service time is exponential with a mean equal to 3 minutes. Suppose the arrival rate is increased by 20%. Determine the change in the average number of customers in the system and the average time a customer spends in the system.
Consider a Markov chain with state space S = {1, 2, 3, 4} and transition matrix...
Consider a Markov chain with state space S = {1, 2, 3, 4} and transition matrix P= where (a) Draw a directed graph that represents the transition matrix for this Markov chain. (b) Compute the following probabilities: P(starting from state 1, the process reaches state 3 in exactly three time steps); P(starting from state 1, the process reaches state 3 in exactly four time steps); P(starting from state 1, the process reaches states higher than state 1 in exactly two...
Consider the Markov chain on state space {1,2, 3,4, 5, 6}
Consider the Markov chain on state space {1,2, 3,4, 5, 6}. From 1 it goes to 2 or 3 equally likely. From 2 it goes back to 2. From 3 it goes to 1, 2, or 4 equally likely. From 4 the chain goes to 5 or 6 equally likely. From 5 it goes to 4 or 6 equally likely. From 6 it goes straight to 5. (a) What are the communicating classes? Which are recurrent and which are transient? What...
Q.4 [8 marks] Consider the Markov chain with the following transition diagram 1 0.5 0.5 0.5 0.5 0.5 2 3 0.5 (a) Write d...
Q.4 [8 marks] Consider the Markov chain with the following transition diagram 1 0.5 0.5 0.5 0.5 0.5 2 3 0.5 (a) Write down the transition matrix of the Markov chain 1 marks 2 marks (b) Compute the two step transition matrix of the Markov chain (c) What is the state distribution T2 for t = 2 if the initial state distribution for 2 marks t 0 is o (0.1, 0.5, 0.4)T? 3 marks (d) What is the average time...
Consider the Markov chain with the following transition diagram. 1 0.5 0.5 0.5 0.5 0.5 2 3 0.5 (a) Write down the trans...
Consider the Markov chain with the following transition diagram. 1 0.5 0.5 0.5 0.5 0.5 2 3 0.5 (a) Write down the transition matrix of the Markov chain (b) Compute the two step transition matrix of the Markov chain 2 if the initial state distribution for 2 marks (c) What is the state distribution T2 for t t 0 is To(0.1, 0.5, 0.4)7? [3 marks (d) What is the average time 1.1 for the chain to return to state 1?...
Consider the M/M/16 queuing system λ=8 μ=14 and p = λ/(sμ) (a) average number of customers in the system (b) average waiting time of each customer who enters the system (c) probability that all se...
Consider the M/M/16 queuing system λ=8 μ=14 and p = λ/(sμ) (a) average number of customers in the system (b) average waiting time of each customer who enters the system (c) probability that all servers are occupied We were unable to transcribe this imageWe were unable to transcribe this imagePU > s) = (s!)(1-p) We were unable to transcribe this image PU > s) = (s!)(1-p)
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ...
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 1/μ. (a) What is the joint p.d.f of X and Y? (b) Set up a double integral for determining Pt <X <Y) (c) Evaluate the above integral. (d) Which of the following equations true, and which are false? {Z > t} = {X > t, Y > t} (e) Compute P[Z> t) wheret 0. (f) Compute the p.d.f. of Z.
1. Consider the M/M/1 queue where the arrive rate is λ and the service rate is μ (a) Give conditions on and such that the stationary distribution exists (b) For the rest of this problem, assume the stationary distribution exists. Calculate the stationary distribution (c) What is the expected number of individuals in the system at a given time? (d) When a new customer arrives into the queue, how long would they be expected to wait until the leave the...
4. Consider an irreducible Markov chain with finite state space S = {0, 1, , (a) Starting at state i, what is the probability that it will ever visit state j? (i,j arbi trary (b) Suppose that Xjj iyi for al i. Let ai P(visit N before 0 start at i). Show uations that the r, satisfy, and show that Xi . H2nt: Derive a system of linear eq that xi- solves these equations
2. (10 points) Consider a continuous-time Markov chain with the transition rate matrix -4 2 2 Q 34 1 5 0 -5 (a) What is the expected amount of time spent in each state? (b) What is the transition probability matrix of the embedded discrete-time Markov chain? (c) Is this continuous-time Markov chain irreducible? (d) Compute the stationary distribution for the continuous-time Markov chain and the em- bedded discrete-time Markov chain and compare the two
2. (10 points) Consider a...
Q.4 [8 marks] Consider the Markov chain with the following transition diagram 1 0.5 0.5 0.5 0.5 0.5 2 3 0.5 (a) Write down the transition matrix of the Markov chain 1 marks 2 marks (b) Compute the two step transition matrix of the Markov chain (c) What is the state distribution T2 for t = 2 if the initial state distribution for 2 marks t 0 is o (0.1, 0.5, 0.4)T? 3 marks (d) What is the average time...
Consider the Markov chain with the following transition diagram. 1 0.5 0.5 0.5 0.5 0.5 2 3 0.5 (a) Write down the transition matrix of the Markov chain (b) Compute the two step transition matrix of the Markov chain 2 if the initial state distribution for 2 marks (c) What is the state distribution T2 for t t 0 is To(0.1, 0.5, 0.4)7? [3 marks (d) What is the average time 1.1 for the chain to return to state 1?...
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 1/μ. (a) What is the joint p.d.f of X and Y? (b) Set up a double integral for determining Pt <X <Y) (c) Evaluate the above integral. (d) Which of the following equations true, and which are false? {Z > t} = {X > t, Y > t} (e) Compute P[Z> t) wheret 0. (f) Compute the p.d.f. of Z.
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