QUEUING MODELS

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QUEUING MODELS

• Queuing theory is the analysis of waiting lines
• It can be used to
• Determine the checkout stands to have open at a
  store
• Determine the type of line to have at a bank
• Determine the seating procedures at a restaurant
• Determine the scheduling of patients at a clinic
• Determine landing procedures at an airport
• Determine the flow through a production process
• Determine the toll booths to have open on a
  bridge

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COMPONENTS OF QUEUING MODELS

• Arrival Process
• Waiting in Line
• Service/Departure Process
• Queue -- The waiting line itself
• System -- All customers in the queuing area
• Those in the queue
• Those being served

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ARRIVAL PROCESS

• Deterministic or Probabilistic (how?)
• Determined by customers in system/balking?
• Single or batch arrivals
• Priority or homogeneous customers

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THE WAITING LINE

• One long line or several smaller lines
• Jockeying allowed?
• Finite or infinite line length
• Customers leave line before service?
• Single or tandem queues

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THE SERVICE PROCESS

• Single or multiple servers
• Deterministic of probabilistic (how?)
• All servers serve at same rate?
• Speed of service depends on line length?
• FIFO/LIFO or some other service priority

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OBJECTIVE

• To design systems that optimize some criteria
• Maximizing total profit
• Minimizing average wait time for customers
• Meeting a desired service level

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TYPICAL SERVICE MEASURES

• Average Number of customers in the system -- L
• Average Number of customers in the queue -- Lq
• Average customer time in the system -- W
• Average customer waiting time in the queue -- Wq
  
• Probability there are n customers in the system
  -- pn
• Average number of busy servers (utilization rate)
  -- ?

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POISSON ARRIVAL PROCESS

• REQUIRED CONDITIONS
• Orderliness
• at most one customer will arrive in any small
  time interval of ?t
• Stationarity
• for time intervals of equal length, the
  probability of n arrivals in the interval is
  constant
• Independence
• the time to the next arrival is independent of
  when the last arrival occurred

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NUMBER OF ARRIVALS IN TIME t

• Assume ? the average number of arrivals per
  hour (THE ARRIVAL RATE)
• For a Poisson process, the probability of k
  arrivals in t hours has the following Poisson
  distribution

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Time Between Arrivals

• The average time between arrivals is 1/?
• For a Poisson process, the time between arrivals
  in hours has the following exponential
  distribution
• f(x) ?e-?t
• This means
• P(next arrival occurs gt t hours from now) e-?t
• P(next arrival occurs within the next t hours)
  1- e-?t

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POISSON SERVICE PROCESS

• REQUIRED CONDITIONS
• Orderliness
• at most one customer will depart in any small
  time interval of ?t
• Stationarity
• for time intervals of equal length, the
  probability of completing n potential services in
  the interval is constant
• Independence
• the time to the completion of a service is
  independent of when it started
• IS THIS A GOOD ASSUMPTION?

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NUMBER OF POTENTIAL SERVICES IN TIME t

• Unlike the arrival process, there must be
  customers in the system to have services
• Assume ? the average number of potential
  services per hour (SERVICE RATE)
• For a Poisson process, the probability of k
  potential services in t hours has the following
  Poisson distribution

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THE SERVICE TIME

• The average service time is 1/?
• For a Poisson process, the service time has the
  following exponential distribution
• f(x) ?e-?t
• This means
• P(the service will take t additional hours)
  e-?t
• P(the remaining service will take longer than t
  hours) 1- e-?t

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TRANSIENT vs. STEADY STATE

• Steady state is the condition that exists after
  the system has been operational for a while and
  wild fluctuations have been smoothed out
• Until steady state occurs the system is in a
  transient state -- transiting to steady state
• It is the long run steady state behavior that we
  will measure

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CONDITIONS FORSTEADY STATE

• For any queuing system to be stable the overall
  arrival rate must be less than the overall
  potential service rate, i.e.
• For one server ? lt ?
• For k servers with the same service rate ? lt
  k?
• For k servers with different service rates
• ? lt ?1 ?2 ?3 ?k

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STEADY STATEPERFORMANCE MEASURES

• Weve mentioned these before
• Average Number of customers in the system -- L
• Average Number of customers in the queue -- Lq
• Average customer time in the system -- W
• Average customer waiting time in the queue -- Wq
  
• Probability there are n customers in the system
  -- pn
• Average number of busy servers (utilization rate)
  - ?

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Littles Laws and Other Relationships

• Littles Laws relate L to W and Lq to Wq by
• L ?W
• Lq ?Wq
• Also, ( in Sys) ( in queue) ( being
  served)
• Thus
• E( in Sys) E( in queue) E( being served)
• L Lq
  ?
• Thus knowing one of L, W, Lq and Wq allows us to
  find the other values.

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CLASSIFICATION OF QUEUING SYSTEMS

• Queuing systems are typically classified using a
  three symbol designation
• (Arrival Dist.)/(Service Dist.)/( servers)
• Designations for Arrival/Service distributions
  include
• M Markovian (Poisson process)
• D Deterministic (Constant)
• G General

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M/M/1

• 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

• Average Number of customers in the system -- L
  ?/(?- ?)
• Average Number of customers in the queue -- Lq
  L - ?/?
• Average customer time in the system -- W L/
  ?
• Average customer waiting time in the queue -- Wq
  Lq/ ?
• Probability 0 customers in the system -- p0
  1-?/?
• Probability n customers in the system -- pn
  (?/?)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 miuntes
• ? (60min./hr)1/12 cust/min. 60/12 5/hr.
• Service times are exponentialand average 8 min.
• ? (service rate) (60min/hr)(1/8cust./min.)
  7.5/hr.
• One server
• This is an M/M/1 system
• Will steady state be reached?
• ? 5 lt ? 7.5/hr. YES

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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. n customers in the system -- pn(?/?)n p0
  (2/3) n(1/3)

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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 could program formulas into EXCEL cells
• WINQSB gives us results

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M/M/k SYSTEMS

• 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

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M/M/k PERFORMANCE MEASURES

• Formulas much more complex e.g.

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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 lt 3(? 40/hr.) i.e. 100 lt 120
• STEADY STATE will be reached

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Solution

• Using WINQSB, with ? 100, ? 40, k 3
• Average system utilization rate ?/k?
  100/120.83
• Avg of busy servers ? ?/? (100/40) 2.5
• 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.
• Prob.0 customers in the system -- p0 .044944

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M/G/1 Systems

• M Customers arrive according to a Poisson
  process at an average rate of ? / hr.
• G Service times have a general distribution
  with an average service time 1/? hours and
  standard deviation of ? hours (1/? and ? in same
  units)
• 1 one server
• Cannot get formulas for pn but can get
  performance measures

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Example -- Teds TV Repair

• Customers arrive according to a Poisson process
  once every 2.5 hours --
• ? 1/2.5 .4/hr.
• Repair times average 2.25 hours with a standard
  deviation of 45 minutes
• ? 1/2.25 .4444/hr.
• ? 45/60 .75 hrs.
• Ted is the only repairman k 1
• THIS IS AN M/G/1 SYSTEM

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FINITE QUEUES

• Frequently there are systems that have limits to
  the maximum number of customers in the system F
• Thus with probability pF the system is FULL and
  an arriving customer cannot join the queue-- i.e.
  we lose pF portion of the potential customers
• Thus the effective arrival rate is ?e 1 - pF
• Use ?e to calculate L, Lq, W, and Wq

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M/M/1 QUEUES WITH FINITE CALLING POPULATIONS

• Maximum m school buses at repair facility, or m
  assigned customers to a salesman, etc.
• 1/? average time between repeat visits for each
  of the m customers
• ? average number of arrivals of each customer
  per time period (day, week, mo. etc.)
• 1/? average service time
• ? average service rate in same time units as ?

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ECONOMIC ANALYSES

• Each problem is different
• To determine the minimum number of servers to
  meet some service criterion (e.g. an average of lt
  4 minutes in the queue) -- trial and error with
  M/M/k systems
• To compare 2 or more situations --
• consider the total (hourly) cost for each system
  and choose the minimum