Math 311

Class 25

Some Fundamental Quantities

  • \(L\): the average number of customers in the system
  • \(L_Q\): the average number of customers in the queue
  • \(L_S\): the average number of customers being served
  • \(W\): the average amount of time a customer spends in the system
  • \(W_Q\): the average amount of time a customer spends in the queue
  • \(W_S\): the average amount of time a customer spends being served
  • \(L = \lambda_a W\)
  • \(L_Q = \lambda_a W_Q\)
  • \(L_S = \lambda_a W_S\)

\(\displaystyle P_n = \lim_{t\to\infty} \operatorname{P}(X(t) = n)\)

Single Server Exponential System (M/M/1)

  • Arrivals are according to a Poisson process with rate \(\lambda_a\).
  • Service times are independent exponential variables with rate \(\lambda_s\).
  • There is one server with one queue.

Process with states \(n = 0, 1, \ldots\), where \(n\) is the number of customers in the system.

Balance equations:

\[ \begin{aligned} \lambda_a P_0 &= \lambda_s P_1\\ (\lambda_a + \lambda_s) P_n &= \lambda_a P_{n-1} + \lambda_s P_{n+1} \text{ for } n > 0\\ \sum_{n=0}^\infty P_n &= 1 \end{aligned} \]

\[P_n = \left(\frac{\lambda_a}{\lambda_s}\right)^n\left(1 - \frac{\lambda_a}{\lambda_s}\right) \text{ for } n = 0, 1, \ldots\]

Fundamental Quantities

\(\displaystyle L = \frac{\lambda_a}{\lambda_s - \lambda_a}\)

\(\displaystyle W = \frac{L}{\lambda_a} = \frac{1}{\lambda_s - \lambda_a}\)

\(\displaystyle W_S = \frac{1}{\lambda_s}\)

\(\displaystyle L_S = \frac{\lambda_a}{\lambda_s}\)

\(\displaystyle W_Q = W - W_S = \frac{\lambda_a}{\lambda_s(\lambda_s - \lambda_a)}\)

\(\displaystyle L_Q = \lambda_a W_Q = \frac{\lambda_a^2}{\lambda_s(\lambda_s - \lambda_a)}\)

Two sequential servers, no queue

  • Two servers (chairs), customer goes to server 1 first, then to server 2.
  • Customers arrive according to a Poisson process with rate \(\lambda\).
  • Arriving customer only enters when server 1 available, otherwise leaves.
  • Serving times of the servers are exponential with rates \(\lambda_1\) and \(\lambda_2\).
  • When a customer is done with server 1 but server 2 is busy, the customer waits at the server 1 station for their turn with server 2.

Balance equations:

\[ \begin{aligned} \lambda P_{00} &= \lambda_2 P_{01}\\ \lambda_1P_{10} &= \lambda P_{00} + \lambda_2 P_{11}\\ (\lambda + \lambda_2)P_{01} &= \lambda_1 P_{10} + \lambda_2 P_{1'1}\\ (\lambda_1 + \lambda_2)P_{11} &= \lambda P_{01}\\ \lambda_2P_{1'1} &= \lambda_1 P_{11}\\ P_{00} + P_{01} + P_{10} + P_{11} + P_{1'1} &= 1 \end{aligned} \]

One server with bulk service

  • Customers arrive “Poissonly” with rate \(\lambda_a\).
  • One server can serve one or two customers at the same exponential rate \(\lambda_s\).
  • If the server is busy, customers join the queue.

Balance equations:

\[ \begin{aligned} \lambda_a P_{0'} &= \lambda_s P_0\\ (\lambda_a + \lambda_s) P_0 &= \lambda_a P_{0'} + \lambda_s (P_{1} + P_2)\\ (\lambda_a + \lambda_s) P_n &= \lambda_a P_{n-1} + \lambda_s P_{n+2} \text{ for } n > 0\\ P_{o'} + \sum_{n=0}^\infty P_n &= 1 \end{aligned} \]