Can you tell me how the redbox is derived? Thank you so much!! Suppose that cust

Question

Can you tell me how the redbox is derived? Thank you so much!!

Suppose that customers arrive at a train depot in accordance with a renewal process having a mean interarrival time mu . Whenever there are N customers waiting in the depot, a train leaves. If the depot incurs a cost at the rate of nc dollars per unit time whenever there are n customers waiting, what is the average cost incurred by the depot? If we say that a cycle is completed whenever a train leaves, then the preceding is a renewal reward process. The expected length of a cycle is the expected time required for N customers to arrive and, since the mean interarrival time is mu , this equals E[length of cycle] = N mu If we let Tn denote the time between the nth and (n+ 1)st arrival in a cycle, then the expected cost of a cycle may be expressed as E[cost of a cycle] = E[c T1 + 2c T2 + . . . + (N - 1)cTN -1 ] which, since E[Tn]= mu , equals Hence, the average cost incurred by the depot is c mu N(N - 1)/2N mu = c(N - 1)/2

Explanation / Answer

E[cost of cycle] = E[cT1 + 2cT2 + ... + (N-1)cTN-1]

Expectations are linear, so we can reduce it as follows:

Factor out c:

E[cost of cycle] = c*E[T1 + 2T2 + ... + (N-1)TN-1]

Break up terms:

= c*{E[T1] + E[2T2] + ... + E[(N-1)TN-1]}

Factor out individual multipliers:

= c*{E[T1] + 2E[T2] + ... + (N-1)E[TN-1]}

Since E[Tn] = , we can substitute that in for all E[Tn], where n = 1, 2, ..., N-1

= c*{ + 2 + ... + (N-1)}

Factor out :

= c*[1 + 2 + ... + (N-1)]

The summation in brackets is a basic summation for a triangular number. It is equivalent to (N-1)N/2:

= c*N*(N-1)/2

And that's the solution!

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.