The exponential distribution is frequently used as a model for the distribution

Question

The exponential distribution is frequently used as a model for the distribution of times between the occurrence of successive events, such as customers arriving at a service facility or calls coming in to a switchboard. Suppose you are working for Manhattan College's IT department and your department's response time (time between receiving an e-mail request 2. IT services and time to setting up an IT ticket) follows an exponential distribution with expected response time equal to 15 seconds. Your supervisor finds this response time unacceptable and requests that your response time decrease from 15 seconds to 10 seconds. Using what you know about probability: a. Provide evidence that his request is reasonable or unreasonable. Provide evidence that 98% of the time your department responds to an IT request within 1 minute. Your supervisor is adamant and wants you to be able to respond to an IT ticket within 10 seconds 95% of the time. What must be your new mean processrateand return period to meet this demand? Is this a reasonable request by your supervisor? c.

Explanation / Answer

Question (2)

(a) Yes, his request is reasonable as response time must be decresed for any IT department or facitlity to reduce the queue and faster servicing the customers.

(b) Here Pr(t < 1minute) = Pr(t < 60 seconds) ~ EXP(t <60 seconds ; mean = 15seconds)

so Here = 1/15 seconds-1

F(t) = 1 - e-t

EXP(t <60 seconds ; mean = 15seconds) = 1 - e-t =1 - e-1/15 * 60 = 1 - 0.1832 = 0.9817

so yes 98% of time response rate is less than 98%

(c) Here the supervisor wants that 95% of the time, i must be able to respond to an IT ticket.

That means, mathematically,

Pr(t < 10 minuts) = 0.95

let say new process rate = seconds-1

so as per exponential cumuative distibution.

EXP(t < 10 minutes ; 1/) = 0.95

1 - e-10 *   = 0.95

e-10 * = 0.05

-10 * = ln (0.05)

1/ = -10/  ln (0.05) = -10/ -3 = 3.33 seconds

= 0.3 seconds-1

The process rate =  0.3 seconds-1

Return period = 3.33 seconds

So, this request by supervisor is totally unreasonable as i have to reduce my mean process time around 5 times. Which is not possible.

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