The Poisson Process Model

1
Example 14.2

• The Poisson Process Model

2
Poisson Process

• The Poisson distribution counts the number of
  events in a certain length of time.
• There is a close relationship between the
  Exponential Distribution and the Poisson
  Distribution.

3
Background Information

• The Smalltown postal branch employs a single
  clerk.
• Customers arrive at this postal branch according
  to a Poisson process at rate 30 customers per
  hour, and the average service time is
  exponentially distributed with mean 1.5 minutes.
• All arriving customers enter the branch,
  regardless of the number already waiting in line.

4
Background Information -- continued

• The manager of the postal branch would ultimately
  like to decide whether to improve the system.
• To do this, she first needs to develop a queuing
  model that describes the steady-state
  characteristics of the current system.

5
Solution

• To begin, we must choose a common unit of time
  and then express the arrival and service rates (?
  and ?) in this unit.
• We could measure time in seconds, minutes, hours,
  or any other convenient time unit, as long as we
  are consistent.
• Here we will choose minutes. Then, because 1
  customer arrives every 2 minutes, ? ½. Also, ?
  0.667.

6
Solution -- continued

• For the M/M/1 model, it turns out that the server
  utilization equals ?/?, the arrival rate divided
  by the service rate.
• To ensure that the system is stable, we must
  require that the server utilization is less than
  1, so that the arrival rate is less than the
  services rate.
• Otherwise, waiting lines will tend to grow
  indefinitely in the long run.
• In general, the formulas for the M/M/1 model are
  somewhat complex. Therefore, we have implemented
  them in an M/M/1 template file.

7
MM1_TEMPLATE.XLS

• This file contains the model.
• The template is shown on the next slide.

8
(No Transcript)
9
Solution -- continued

• We will not provide step-by-step instructions
  because we expect that you will use this as a
  template rather than enter the formulas yourself.
• However, we make the following points.
• All you need to enter are the inputs in B4
  through B6.
• You can enter numbers for the rates in cells B5
  and B6, or you can base these on observed data.
  The data sheet exists in the template and can be
  seen on the next slide. Given such data, you
  would enter the formulas 1/Data!B5 and
  1/Data!C5 in cells B5 and B6 of the Template
  sheet.

10
(No Transcript)
11
Solution -- continued

• The values in B5, B15 and B17 are related by the
  equation version of Littles formula.
• The template is set up so that you can enter any
  value for the n in cell E11 to obtain the
  steady-state probability of n customers in the
  system in cell F11. You can even enter several
  values of n in cells E11, E12 and so on, and then
  copy the formula in cell F11 down for these
  values. Similarly, you can enter any time t in
  cell H11 and obtain the probability that a
  typical customer will wait in the queue at least
  time t in cell I11. The values of n and t shown
  are for illustration only.

12
Solution -- continued

• From the template we see, for example, that when
  the arrival rate is 0.5 and the service rate is
  0.667, the expected number of customers in the
  queue is 2.25 and the expected time a typical
  customer spends in the queue is 4.5 minutes.
• However 25 of all customers spend no time in the
  queue, while 53.7 spend more than 2 minutes in
  the queue.
• Also, we see that the steady probability of
  having exactly 4 customers in the system is
  0.079. Equivalently, there are exactly 4
  customers in the system 7.9 of the time.

13
Solution -- continued

• The bank manager can experiment with other
  arrival rates or services rates in cells B5 and
  B6 to see how the various output measures are
  affected.
• One particularly important insight can be
  obtained through a data table.
• The current server utilization is 0.75, and the
  system is behaving fairly well, with short waits
  in queue on average.
• The data table, however, shows how bad things can
  get when the service rate is just barely above
  the arrival rate, so that the server utilization
  is just barley below 1.

14
Solution -- continued

• The corresponding line chart shows that the
  expected time in queue increases extremely
  rapidly as the service rate approaches the
  arrival rate.
• Whatever else the bank manager learns from this
  model, she now knows that she does not want a
  services rate close to the arrival rate, at least
  not for extended periods of time.