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Queuing Models

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Queuing Models M/M/k Systems * * * * * * * CLASSIFICATION OF QUEUING SYSTEMS Recall that queues are classified by (Arrival Dist.)/(Service Dist.)/(# servers ... – PowerPoint PPT presentation

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Title: Queuing Models


1
Queuing Models
  • M/M/k Systems

2
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.

3
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

4
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
  • ? utilization rate or Average number customers
    being served ?/?

5
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 lt ? 7.5/hr. YES

6
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

7
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

8
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9
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 ? lt kµ

10
M/M/k PERFORMANCE MEASURES
11
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 ? lt 3µ, i.e. 100 lt 120,
  • STEADY STATE will be reached

12
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.
  • Average system utilization rate ? ?/k?
    100/120 .83
  • Avg of busy servers k? ?/? (3X0.83) 2.5

13
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14
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 achieved regardless of the
    values of ? and µ!
  • Formulas for steady state quantities are complex
    use template.

15
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 lt F customers in
    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).

16
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.

17
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 gt 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)
18
Effective Arrival Rate ?e ?(1-pF) C4(1-P12)
Excel 10(1-.06667) 9.3333
19
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/k model
  • Finite queuing models
  • Always reach steady state
  • Effective arrival rate, ?e ?(1-pF)
  • Use of Templates
  • M/M/k
  • M/M/k/F
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