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In: Advanced Math

The emergency room at Hosptal Systems, Inc. (HIS) serves patients who arrive according to a Poisson...

The emergency room at Hosptal Systems, Inc. (HIS) serves patients who arrive according to a Poisson distribution at the rate of 9 per hour. Treatment takes an average of 6 minutes and the treatment times can be considered to follow an exponential distribution. What is the (a) minimum number of doctors required so that at least 70% of the arriving patients can receive treatment immediately? (b) minimum number of doctors required so that the average time a patient waits for treatment is no more than 30 minutes as advertised? No more than 15 minutes?

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Expert Solution

IF YOU HAVE ANY DOUBTS COMMENT BELOW I WILL BE TTHERE TO HELP YOU..ALL THE BEST..

AS FOR GIVEN DATA..

The emergency room at Hosptal Systems, Inc. (HIS) serves patients who arrive according to a Poisson distribution at the rate of 9 per hour. Treatment takes an average of 6 minutes and the treatment times can be considered to follow an exponential distribution. What is the

EXPLANATION ::

(a) minimum number of doctors required so that at least 70% of the arriving patients can receive treatment immediately?

SOL ::

(a)

Arrival rate = 9 per hour

Service rate = 60/6 = 10 per hour

For 70% of the patients to receive treatment immediately, utilization rate must be at most = 1 - 70% = 30%

Therefore, minimum number of doctors required = Arrival rate / (Service rate * utilization rate)

= 9/(10*30%)

= 3 doctors

(b) minimum number of doctors required so that the average time a patient waits for treatment is no more than 30 minutes as advertised? No more than 15 minutes?

SOL ::

With multiple doctors, this is an M/M/s queue system

Operating characteristics are computed using the following spreadsheet

For target waiting time (Wq) <= 30 minutes or 0.5 hour, Look for Wq less than 0.5 in column I of the above spreadsheet.

The value of Wq just less than 0.5 is 0.0254 (=1.52 minutes) . Corresponding value of s = 2

So, 2 doctors are needed. Average waiting time is 1.52 minutes which is less than 30 minutes as well as 15 minutes.

So, a minimum of 2 doctors are needed.

I HOPE YOU UNDERSTAND..

PLS RATE THUMBS UP..ITS HELPS ME ALOT..

THANK YOU...!!

Colby Messinger answered 1 year ago
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