Question

There are 3 servers in the checkout area. The interarrival time of customers is 2 minutes. The processing time is 5 minutes. The coefficients of variation for the arrival process and the service process are 1 and 0.85 respectively. What will be the impact on the average number of customers in service if one more server is added

Answer 1

Answer:

As the one more server is added the Average number of customers in service reduces from  ( 5.585 to 3.019  )    which is a reduction of 2.566

Explanation:

interval time of customers ( a )  = 2 minutes

processing time ( p ) = 5 minutes

coefficient of variation for arrival process ( Cva ) = 1

coefficient of variation for service process ( Cvp ) = 0.85

Determine the impact on the average number of customers in service if one more server is added

For three (3) servers (  m ) we will have

flow rate = 1 / a = 1/2 = 0.5

x =  ( 2 (m+1) -1 ) ^0.5

where m = 3 ( number of servers )

hence X = 2.645

now lets determine

i) utilization =  p / (m*a). where p = 5 , m = 3 , a = 2

hence utilization ( u ) = 0.833 ,  1 - u = 1 - 0.833 = 0.167

ii) Time spent in queue

time spent in queue = 6.17 minutes

number of customers been served = mu = 3 * 0.833 = 2.500

number of customers waiting to be served = waiting time / arrival time

= 6.17 / 2 = 3.085

iii) Average customer in queue = number of customers been serviced + number of customers waiting to be served =  2.5 + 3.085 = 5.585

For 4 servers  ( m = 4 )

x = ( 2 (m+1) -1 ) ^0.5  

   = ( 2 ( 5 ) - 1 ) ^ 0.5 = 3

utilization =  p / (m*a). where p = 5 , m = 4 , a = 2

   hence utilization = 0.625 ,  

Time spent in queue =  1.039 minutes  ( using equation 2 )

number of customers been serviced = m* u = 2.5

number of customers waiting to be served = waiting time / arrival time

= 1.039 / 2 = 0.520

Average customer in queue = number of customers been serviced + number of customers waiting to be served = 2.5 + 0.520 = 3.019

hence as the one more server is added the Average number of customers in service reduces from  ( 5.585 to 3.019  )    which is a reduction of 2.566