$($

$2$

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$$Figure

$1$

$4$

$.$

$8$

$$shows a

$2$

$-$

server system with a waiting room that can hold only

$1$

$$job$.$ Any arrival that sees

$3$

$$jobs already in the system is dropped. Jobs arrive from outside according to a Poisson process with rate

$$

$=$

$1$

$.$

$$Whenever a server fifinishes serving a job, it grabs the job from the waiting area, if there is one. Job sizes are Exponentially distributed with rate

$$

$=$

$1$

$.$

$($

a

$)$

$$Draw a CTMC where the state represents the total number of jobs in the system.

$($

b

$)$

$$Suppose that there are exactly

$2$

$$jobs in the system. What is the probability that a job arrives before a job completes?

$($

c

$)$

$$Use your CTMC to determine the probability that the system is idle

$($

both servers are idle

$)$

$.$

$($

d

$)$

$$What is the throughput of the system?

$($

e

$)$

$$What is

$,$

$$the expected number of jobs in the system?

$($

f

$)$

$$What is

$$

$[$

$7$

$]$

$,$

$$the expected response time

$($

for those jobs not dropped

$)$

$?$

$($

g

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$$Consider the process of arrivals to the system that are not dropped. Is this a Poisson process? Why or why not? Show Image Transcript