Suppose you call your cell phone company to address a problem with your service.

Question

Suppose you call your cell phone company to address a problem with your service. When you call, it takes 1 minute to navigate through the menus before you can request a live operator. Your wait time for a live operator is exp(0.5) distributed. Once you get a live operator, one of two things happens. With probability 0.2, you get a competent operator who can resolve your problem in time that is exp(0.2). With probability 0.8, you get an incompetent operator who gives you the run-around and eventually returns you to the queue for a live operator, after wasting your time with distribution exp(0.1). What is the expected time before your problem is resolved?

Explanation / Answer

Expected time before your problem resolved.

Time to navigate the menu =1 minute

The expected value of the Exponential() distribution = 1/

Wait time for a live operator is exp(0.5) distributed i.e

Expected wait time for a live operator = 1/0.5 = 2minutes.

Probability of getting a competent operator = 0.2

Probability of getting an incompetent operator = 0.8

competent operator resolution time is exp(0.2)

Competent Expected resolution time = 1/0.2 = 5

Incompetent operator : Wasting time is exp(0.1)

Incompetent operator :Expected waiting time = 1/0.1 = 10

Expected resolution time = Probability of getting a competent operator x Competent - Expected resolution time + Probability of getting a incompetent operator x Incompetent operator expected waiting time = 0.2x5+0.8x10 =1+8=9

Expected time before your problem is resolved = Time to navigate the menu + Expected wait time for a live operator +Expected resolution time = 1+2+9 = 12 minutes

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