Queuing Models

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Queuing Models

• M/M/k Systems

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CLASSIFICATION OF QUEUING SYSTEMS

• Recall that queues are classified by
• (Arrival Dist.)/(Service Dist.)/( servers)
• Designations for Arrival/Service distributions
  include
• M Markovian (Poisson process)
• D Deterministic (Constant)
• G General
• We begin with the basic model, the M/M/1 system.

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M/M/1

• An M/M/1 system is one with
• M Customers arrive according to a Poisson
  process at an average rate of ?/hr.
• M Service times have an exponential
  distribution with an average service time 1/?
  hours
• 1 one server
• Simplest system -- like EOQ for inventory -- a
  good starting point

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M/M/1PERFORMANCE MEASURES

• For the M/M1 system the performance measures are
  given by these simple formulas
• L Average of customers in the system ?/(?-
  ?)
• LQ Average of customers in the queue L -
  ?/?
• W Average customer time in the system L/ ?
• WQ Average customer time in the queue Lq/ ?
• p0 Probability 0 customers in the system
  1-?/?
• pn Probability n customers in the system
  (?/?)n p0
• ? Average number of busy servers (utilization
  rate) or Average number customers being served
  ?/?

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EXAMPLE -- Marys Shoes

• Customers arrive according to a Poisson Process
  about once every 12 minutes
• Service times are exponential and average 8 min.
• One server
• This is an M/M/1 system with
• ? (60min./hr)/(12 min./customer) 60/12
  5/hr.
• ? (service rate) (60min/hr)/(8min./customer)
  7.5/hr.
• Will steady state be reached?
• ? 5

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MARYS SHOESPERFORMANCE MEASURES

• Avg of busy servers (utilization rate) or
• Avg customers being served ? ?/? (5/7.5)
  2/3
• Average in the system -- L ?/(?- ?)
  5/(7.5-5) 2
• Average in the queue -- Lq L - ?/? 2 -
  (2/3) 4/3
• Avg. customer time in the system -- W L/ ?
  2/5 hrs.
• Avg cust.time in the queue - Wq Lq/ ? (4/3)/5
  4/15 hrs.
• Prob. 0 customers in the system -- p0 1-?/?
  1-(2/3) 1/3
• Prob. 3 customers in the system -- pn(?/?)3 p0
  (2/3)3(1/3)
• 8/81

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COMPUTER SOLUTION

• The formulas for an M/M/1 are very simple, but
  those for other models can be quite complex
• We can use a queuing template to calculate the
  steady state quantities for any number of
  servers, k
• For the M/M/1 model use the M/M/k worksheet in
  Queue Template
• Since k 1, the results are in the row
  corresponding to 1 server

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M/M/k SYSTEMS

• An M/M/k system is one with
• M Customers arrive according to a Poisson
  process at an average rate of ? / hr.
• M Service times have an exponential
  distribution with an average service time 1/?
  hours regardless of the server
• k k IDENTICAL servers
• To reach steady state ?

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M/M/k PERFORMANCE MEASURES
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EXAMPLELITTLETOWN POST OFFICE

• Between 9AM and 1PM on Saturdays
• Average of 100 cust. per hour arrive according to
  a Poisson process -- ? 100/hr.
• Service times exponential average service time
  1.5 min. -- ? 60/1.5 40/hr.
• 3 clerks k 3
• This is an M/M/3 system
• ? 100/hr
• ? 40/hr.
• Since ?
• STEADY STATE will be reached

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Solution

• Using the formulas, with ? 100, ? 40, k 3,
  it can be shown that
• Prob.0 customers in the system -- p0 .044944
• Average in the system -- L 6.0112
• Average in the queue -- Lq 3.5112
• Avg. customer time in the system -- W .0601
  hrs.
• Avg cust.time in the queue - Wq .0351hrs.
• Avg of busy servers ? ?/? (100/40) 2.5
• Average system utilization rate ?/k ?/k?
  100/120 .83

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M/M/k/F Systems

• An M/M/k/F system is one with
• M Customers arrive according to a Poisson
  process at an average rate of ? / hr.
• M Service times have an exponential
  distribution with an average service time 1/?
  hours regardless of the server
• k k IDENTICAL servers
• F maximum number of customers that can be in
  the system at any time
• Because the queue cannot build up indefinitely,
  steady state will be acchieved regardless of the
  values of ? and µ!
• Formulas for steady state quantities are complex
  use template.

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Basic Concept of M/M/k/F Systems

• The number of customers that can be in the system
  is 0, 1, 2, ,F
• If an arriving customer finds the system when he arrives, he will join the
  system.
• If an arriving customer finds F customers in the
  system when he arrives, he cannot join the
  system, he will leave, and his service is lost.
• Thus the effective arrival rate, ?e, the average
  number of arrivals per hour that actually join
  the system is ?e ?(1-pF).

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EXAMPLERYANS ROOFING

• The average number of customers that call the
  company per hour is 10.
• There is 1 operator who averages 3 minutes per
  call.
• Both calls and operator time conform to Poisson
  processes.
• There are 3 phone lines so 2 calls could be on
  hold. A caller that calls when all 3 lines are
  busy, gets the busy signal and does not join the
  system.
• This is an M/M/1/3 system with
• ? 10/hr.
• µ 60/3 20/hr.

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USING THE M/M/k/F TEMPLATE

• The template is designed to be used for the case
  where a queue is possible that is the maximum
  number of customers in the system is greater than
  the number of servers, i.e. F k
• To determine the effective arrival rate, we find
  pF on the right side of the output. Then in a
  cell (or by hand) we can calculate

Effective Arrival Rate ?e ?(1-pF)
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Effective Arrival Rate ?e ?(1-pF) C4(1-P12)
Excel 10(1-.06667) 9.3333
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Review

• M/M/k systems are ones with
• a Poisson arrival distribution
• an exponential service distribution
• k identical servers
• Steady state formulas for M/M/1 model
• Finite queuing models
• Always reach steady state
• Effective arrival rate, ?e ?(1-pF)
• Use of Templates
• M/M/k
• M/M/k/F