Capacity Planning

1
Capacity Planning

• Chapter 14

2
Capacity Planning

• Capacity planning decisions involve trade-offs
  between the cost of providing a service (i.e.,
  increasing the number of servers) and the cost or
  inconvenience to the customer
• Involves determining the appropriate level of
  service capacity by specifying the proper mix of
  facilities, equipment, and labor required to meet
  anticipated demand

3
Classification of Queuing Models
4
Single Server M/M/1

• ? OOOO
• ? arrivals per time period
• O customer
• server

5
M/M/1 Boat Launch Example

• If ? 6 boats arrive per hour
• µ 10 boats launched per hour
• p ? / µ
• Then
• probability that the customer must wait is
• p 6 / 10 0.6 or 60
• probability that the customer does not have to
  wait is
• p0 1- 0.6 0.4 (40)

6
M/M/1 Boat Launch Example

• Mean number of boats in the system
• Ls ? / (µ ?)
• Ls 6 / (10 6) 6/4 1.5 boats
• Mean number of boats in the queue
• Lq (p?) / ((µ ?)
• Lq (.6 x 6) / (10 6) 3.6 / 4 .9 boats

7
M/M/1 Boat Launch Example

• Mean time in the system
• Ws 1 / (µ ?) 1 / (10 6) ¼ hour
• Mean time in queue
• Wq p / (µ ?) 0.6 / (10 6) .15 hour

8
Multiple Servers M/M/c

• ? OOOO
• ? arrivals per time period
• O customer
• servers

9
M/M/4 Secretarial Pool Example

• If ? 9 requests per hour
• c 4 servers
• p ? / c (used to calculate the value of p0)
• Then
• probability that a faculty member does not have
  to wait for secretarial work is
• P0 1
• (9/4)0 (9/4)1 (9/4)2 (9/4)3 (9/4)4
• 0! 1! 2! 3!
  4!(1-9/16)
• P0 0.098

10
M/M/4 Secretarial Pool Example

• Mean number of jobs in the system
• p c1
• Ls (P0 p)
• (c-1)!(c-p)2
• (9/4)5
• Ls
  (0.098) 9/4
• (4-1)!(4-9/4)2
• Ls 2.56

11
M/M/4 Secretarial Pool Example

• Mean time for each secretarial job in the system
• Ls
• Ws
• ?
• 2.56
• Ws 0.28 or 17 minutes per job
• 9