a Formulate this as a Birth and Death process write balance equations and

A formulate this as a birth and death process write

This preview shows page 2 - 4 out of 4 pages.

(a) Formulate this as a Birth and Death process, write balance equations and calculate limiting probabilities. Let X denote the number of customers in the system. Then the state space S = { 0 , 1 , 2 , 3 , 4 } . Let q ( x, y ) denote the exponential rate at which the CTMC moves to state y while in state x . Then q (0 , 1) = q (1 , 2) = q (2 , 3) = q (3 , 4) = 8 , and q (1 , 0) = 2 , q (2 , 1) = 4 , q (3 , 2) = q (4 , 3) = 6 , where all rates have units of services per hour. q ( x, y ) = 0 otherwise. We can now write down our balance equations: p 0 q (0 , 1) = p 1 q (1 , 0) p 1 = 4 p 0 , p 1 ( q (1 , 0) + q (1 , 2)) = p 0 q (0 , 1) + p 2 q (2 , 1) p 2 = 8 p 0 ,
Image of page 2

Subscribe to view the full document.

p 2 ( q (2 , 1) + q (2 , 3)) = p 1 q (1 , 2) + p 3 q (3 , 2) p 3 = 32 / 3 p 0 , , p 3 ( q (3 , 2) + q (3 , 4)) = p 2 q (2 , 3) + p 4 q (4 , 3) p 4 = 128 / 9 p 0 , 4 X i =0 p i = 1 . Substitution into the last constraint yields our stationary distribution: p = (9 / 341 , 36 / 341 , 72 / 341 , 96 / 341 , 128 / 341) = (0 . 02639 , 0 . 10557 , 0 . 21114 , 0 . 28152 , 0 . 37537) . (b) What fraction of potential customers will not enter the system? Note that customers are turned away only when the queue is full, which is when the CTMC is in state 4. By PASTA (Poisson arrivals see time averages) the fraction of potential customers that are turned away is p 4 = 128 / 341 = 0 . 375. (c) What proportion of time is at least one server busy? This is simply 1 - p 0 = 1 - 9 / 341 = 332 / 341 = 0 . 9736. (d) What is the average number of customers in the system? L = 4 X i =0 ip i = 980 / 341 = 2 . 8739 . (e) What is the average waiting time (before the service) of a customer? By Little’s formula we have that the average time W spent by a customer in the queue is equal L/λ a , where λ a = λ (1 - p 4 ) = 8(1 - 0 . 3754) = 4 . 9968. Hence W = 2 . 8739 / 4 . 9968 = 0 . 5751 (hour), and W q = W - E [ S ] = 0 . 5751 - 0 . 5 = 0 . 0751 (hour), or W q = 4 . 5 minutes. 4. [20] consider a taxi station where taxis and customers arrive in accordance with Poisson processes with respective rates of one and two per minute. There is place for 4 taxies at the station, so if there are already 4 taxies at the station a new taxi will not enter. On the
Image of page 3
Image of page 4
  • Spring '08
  • Brown

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes