– planes the circle before they can land
! Example questions:
 – 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?
! arrival process (l  and distribution of interarrival times)
! service process (mand distribution of service times)
! number of servers
 
C the number of parallel servers
N the system capacity
abbreviations for distribution functions:
an infinite target group. The arrival intervals and the service
times are distributed exponentially
 
 
! 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
 
! Utilisation rate (server utilization, percentage of the
time that a server is busy, where c=the number of 
parallel servers)
and queue)
!  Average time spent by a customer in the system w 
(service and queue)
 
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# 
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
 
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)
! 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
time
process)
exponentially, or Erlang distribution
! M/M/1, M/G/1, M/Ek/1, M/D/1
! formulas for M/M/1 system can be derived quite easily
(not material for the exam)
 
! if r  > 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
! Little’s equation: L = l  * w
 
steady-state condition
! But also: P0 * l  + P2 * m= P1 * (l  + m)
! Pn = l n/mn* P0
  0 1 2 3
P3* l
P4* m
! the relation between Pn and P0 is:
! if the system is in steady-state then l /m<1 and:
! can be replaced by:
! ! "=
=
"=
=
## $
% && '
( ==##
$
% && '
! because for L we know that:
! writing it out yields:
! fill in and use Little’s equation. Then we get a table with
the most important values for a M/M/1 system.
n
"
# $$ %
& !! "
# $$ %
& '== ((
)=
=
)=
of customers per minute Poisson distributed)
!  Average service time is 40 seconds per customer 
(exponentially distributed)
! What is:
 – probability there are exactly 5 customers in the
system?
! service times distribution have an unknown standard deviation s 2
! steady state parameters for M/M/1 can be calculated by substituting $
2=1/µ2
! service times distribution is Erlang of order k
k exponential distributions after another with average 1/mand standard deviation 1/km2
! steady state parameters for M/M/1 can be calculated from M/G/1 by substituting $
2=1/kµ2
! blood-test for patients
blood, discussion with doctor 
arrival per hour (Poisson)
 
! arrival times are exponentially distributed
! service times have no variance
steady state parameters for M/D/1 can be calculated from M/G/1 by substituting $
2=0
! Note: the average queue length for M/D/1 is exactly half 
of M/M/1
! one runway
! fuel costs f. 5000,- per hour 
! calculate:
 – expected total number in system
 
system
0 0 1
M / G/ 1 S ta n da r dd = 1 / 1 5
)1(2
0 1 3
M / G/ 1 S ta n da r dd = 1 / 1 0
)1(2
0 2 2
M / G/ 1 St a n da r dd = 1 /5
)1(2
! reduce service time
 
customer leaves
called a
intensity of the entities
!   l e < l
! Parameters: see table
(1 a )(1 a)
P
! 3 chairs to wait
! calculate:
 
! more servers
! if umber of customers in system n < c then new arrival
can be served immediately
! utilisation rate is not r  = l /mbut r  = l /cm
! if r  > 1 then system grows with (l -cm)
! complex formulas, so often tables or graphs are used
 
! arrival intensity 2 per minute (Poisson)
! service time 40 seconds (exponential)
! determine:
 – probability that there are no customers in the system
with the calculated number of servers
 – average queue length with the calculated number of 
servers