april 07 # 11

[MindFull]

An airline has N adjacent check-in desks at a particular airport, each of which can handle any customer from that airline. Arrivals of passengers at the check-in area are assumed to follow a Poisson process with rate q. The time taken to check-in a passenger is assumed to follow an exponential distribution with mean 1/a.

Answer: The possible states are 0 to N desks in use with no passengers queuing, and N desks in use with 0, 1, 2, ….. passengers in the queue.
When all desks are occupied and there are M passengers in the queue denote
the state as N:M.
State space is:
{0, 1, 2, …., N - 1, N : 0, N : 1, N : 2, …..}

I understand why the trans. rate from 1 to 0 is "a" (that's the parameter for the exp func.) but the ques is why do we multiply by the number of desks that are occupied, eg. 2a, Na.

Thanks.

 

[LondonStudent]

We multiply by the number of desks because the probability of someone leaving a desk increases the more desks that are currently taken.

 

[Lewin]

confused

part iv) kolmogorov foward eqtns in component form
The 2nd answer gives d/dt Pr(t)=a(r+1)Pr+1(t)+qPr-1(t)-(ar+q)Pr(t)
for r+1<=N

However,taking say r=2, we can move as follows:
3-3-2
3-2-2
1-1-2
1-2-2

so im getting
d/dt Pr(t)= a(r+1)Pr+1(t)+qPr-1(t)-2(ar+q)Pr(t)

mine has a 2 before the last part while the examiners answer doesnt..........anyone explain where im going wrong?

 

[Hemant Rupani]

part iv) kolmogorov foward eqtns in component form
The 2nd answer gives d/dt Pr(t)=a(r+1)Pr+1(t)+qPr-1(t)-(ar+q)Pr(t)
for r+1<=N

However,taking say r=2, we can move as follows:
3-3-2
3-2-2
1-1-2
1-2-2

so im getting
d/dt Pr(t)= a(r+1)Pr+1(t)+qPr-1(t)-2(ar+q)Pr(t)

mine has a 2 before the last part while the examiners answer doesnt..........anyone explain where im going wrong?

I would do as (transition force in - transition force out)
And get same result....
\( 3aP_3(t)+qP_1(t)-(2a+q)P_2(t) \)
I can't understand how you did?

 

[Lewin]

I figured out my mistake,thanks Mr Rupani