SCM $401$ Queueing Analysis Review Spring $2024$

Independent vs$.$ Pooled vs$.$Super Server$$

The local credit union has a call center staffed with $2$ customer service representatives. Calls from

customers arrive following a Poisson process with an average rate of $12$ calls per hour. The service

times are exponentially distributed, and each representative takes an average of $8$ minutes per

caller. The credit union is considering three different configurations for the call center:

$1.$ Independent Servers: Incoming calls will be equally distributed between the two

representatives on arrival. Each caller will have to wait in line until their designated

representative finishes previously assigned calls, even if the other representative is idle.

$2.$ Pooled Servers: Incoming calls are routed to the first available representative on a first

come first serve basis.

$3.$Super Server$$: The two representatives handle every call together on a first come first

serve basis. Service times are still exponential, but since the servers are sharing the work,

each call only takes $4$ minutes on average.

In each case, calculate the:

a$.$ Average time callers spend on hold $($waiting for representatives$)$

b$.$ Average total time of a call $($including waiting for and talking to representatives$)$

c$.$ Average number of callers on hold $($waiting for a representative$)$

d$.$ Average numbers of callers on call $($including waiting for and talking to representatives$)$