Queuing theory

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Queuing theory

• Examples are
• waiting to pay in the supermarket
• waiting at the telephone for information
• planes the circle before they can land
• Example questions
• what is the average waiting time of a customer?
• how many customers are waiting on average?
• how long is the average service time?
• what is the chance that one of the servers has
  nothing to do?

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Queuing system
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Parameters of queuing systems

• The behaviour of a queuing system is dependent
  on
• arrival process (l and distribution of
  interarrival times)
• service process (m and distribution of service
  times)
• number of servers
• capacity of the system
• size of the population for this system

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Kendalls notation

• A/B/c/N/K where
• A the interarrival distribution
• B the service time distribution
• C the number of parallel servers
• N the system capacity
• K the size of the target group.
• abbreviations for distribution functions
• M exponential
• D constant or deterministic
• Ek Erlang
• G General

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Kendalls notation

• Example
• M/M/1/?/?
• single-server system with unlimited queuing
  capacity and an infinite target group. The
  arrival intervals and the service times are
  distributed exponentially
• If N and K are infinite, they can be left out of
  the notation. M/M/1/?/? is abbreviated to M/M/1

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Kendalls notation

• Exercises
• hairdresser with 3 chairs for a haircut and 5
  waiting chairs
• 6 machines that need to be serviced and 1 service
  engineer with Poisson distributed service time
• planes that land on one airstrip
• queue in a canteen with exponentially distributed
  interarrival times and constant service times

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Output variables

• Utilisation rate ? (server utilization,
  percentage of the time that a server is busy,
  where cthe number of parallel servers)
• Probability of n customers in the system Pn
• Average number of customers in the system L
  (service and queue)
• Average number of customers in the queue Lq
• Average time spent by a customer in the system w
  (service and queue)
• Average time spent by a customer in the queue wq

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Transient versus steady-state behaviour

• transient behaviour (from t0)performance
  indicators such as average waiting time, average
  number of customers in queue, etc. are dependent
  of the time, e.g. wq(t), Lq(t)
• steady-state (stationary) behaviour (t
  ???)performance indicators such as average
  waiting time are not dependent of the time
  anymore the probability that the system is in a
  certain state is completely independent of time,
  e.g. wq, Lq

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Transient behaviour

• history of number of customers in the system
  graph of number of customers versus time
• important to know the queuing strategy
• FIFO (first in first out)
• LIFO (last in first out stack)
• SIRO (service in random order)
• SPT (shortest processing time first)
• PR (priority)

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History hospital exercise

• total number of visits by the one nurse (N)
• for all N visits the starting time
• for all visits the time in system by the patient
• for all visits the time in queue by the patient

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Markov models

• arrival times are exponentially distributed
  (Poisson process)
• formulas for steady-state situation
• service times are drawn from a deterministic,
  exponentially, or Erlang distribution
• M/M/1, M/G/1, M/Ek/1, M/D/1
• M/M/1/N, M/M/c, M/M/1/K/K
• formulas for e.g. Pn, L, Lq, w, wq
• formulas for M/M/1 system can be derived quite
  easily (not material for the exam)

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Utilisation of server

• utilisation of M/../1 is rl/m
• utilisation of M/../c is rl/cm
• if r gt 1then the system is instable on average,
  more customers arrive than the system can handle
• a traffic intensity a is used for systems with
  a finite population

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Littles equation

• system in stationary condition
• number of customers in system is L
• average total time in system is w
• Littles equation L l w

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Markov model M/M/1

• Average number of customers in system for M/M/1
  in steady-state condition
• Steady state, so P0 l P1 m
• But also P0 l P2 m P1 (l m)
• Pn ln/mnP0

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Markov model M/M/1 (cont.)

• for the sum of P, the following equation holds
• the relation between Pn and P0 is
• if the system is in steady-state then l/mlt1 and
• can be replaced by
• Using this, we can calculate that P0 equals

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Markov model M/M/1 (cont.)

• Thus we get for Pn
• because for L we know that
• writing it out yields
• fill in and use Littles equation. Then we get a
  table with the most important values for a M/M/1
  system.

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Markov model M/M/1 (cont.)
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Exercise canteen with one check-out point

• On average, one customer per minute arrives
  (number of customers per minute Poisson
  distributed)
• Average service time is 40 seconds per customer
  (exponentially distributed)
• What is
• average time in system?
• Average waiting time?
• Average number of customers in the queue?
• probability there are exactly 5 customers in the
  system?

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The M/G/1 system

• Arrival times are exponentially distributed
• service times distribution have an unknown
  standard deviation s2
• steady state parameters for M/M/1 can be
  calculated by substituting ?21/?2

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The M/Ek/1 system

• arrival times are exponentially distributed
• service times distribution is Erlang of order kk
  exponential distributions after another with
  average 1/m and standard deviation 1/km2
• steady state parameters for M/M/1 can be
  calculated from M/G/1 by substituting ?21/k?2

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The M/Ek/1 system exercise

• blood-test for patients
• 3 separate doctors or assistants blood pressure,
  taking blood, discussion with doctor
• on average 15 minutes per activity (exponential)
  and 1 arrival per hour (Poisson)
• calculate
• average number of patients in the waiting room

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The M/D/1 system

• arrival times are exponentially distributed
• service times have no variancesteady state
  parameters for M/D/1 can be calculated from M/G/1
  by substituting ?20
• Note the average queue length for M/D/1 is
  exactly half of M/M/1

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The M/D/1 system (exercise)

• planes that land on a runway
• one runway
• 30 arrivals per hour (Poisson)
• 90 seconds deterministic landing time
• fuel costs f. 5000,- per hour
• calculate
• average length of queue
• average waiting time
• expected total number in system
• fuel costs per hour as a result of delay in queue

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Relation between queuing systems
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Reducing waiting times

• reduce number of arrivals per time unit
• reduce service time
• increase number of servers
• decrease variance in service time

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The M/M/1/N system

• if the system is full, the customer leaves
• in this case, l/m is called a
• le is the arrival intensity of the entities that
  stay in the system
• le lt l
• calculation le l(1-PN)
• Parameters see table

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The M/M/1/N system exercise

• Small travel agent with one employee
• 3 chairs to wait
• l 5 customers per hour (Poisson)
• m 10 customers per hour (Poisson)
• calculate
• the effective arrival intensity le

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The M/M/c system

• more servers
• if umber of customers in system n lt c then new
  arrival can be served immediately
• stable system if l lt cm
• utilisation rate is not r l/m but r l/cm
• if r gt 1 then system grows with (l-cm)
• complex formulas, so often tables or graphs are
  used

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The M/M/c system
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The M/M/c system
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The M/M/c system (exercise)

• customers at post office
• 1 queue and more servers
• arrival intensity 2 per minute (Poisson)
• service time 40 seconds (exponential)
• determine
• number of servers needed to reach steady state
• probability that there are no customers in the
  system with the calculated number of servers
• average queue length with the calculated number
  of servers