10
Queuing Theory
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Queuing theory is concerned with analysis of those situations where customers seek some
service, for which they arrive at a service station, get the service and then leave.
Customers calling at bank tellers, reservations, at railway counters, sale of tickets at a
theatre, flow of ships to seashore, occurrences of fires etc. illustrate such situations. The
general structure of a queuing system has the following features: arrival of customers
from some source population, waiting in the queue, entering the service station for
obtaining the desired service and, finally, leaving the system. The elements of a system
are given below:
1. Arrival process: The arrival of customers, who may be people, aeroplanes, ships,
machines, or any other inanimate object, may be classified on different bases:
(a) According to source: The source of population customers may be large (even
infinite) or small. Customers boarding buses represent a large population
while machines in a plant, which require repairs from time to time, exemplify
small population.
(b) According to numbers: The customers may arrive in single numbers or in
groups.
(c) According to time: arrival of customers can be at known times, like patients
visiting a specialist doctor, or, more generally, at random points in time. The
queuing models mostly are based on the assumption of random arrival of
customers.
2. Service system: One aspect of a service system is its structure and the other is speed
of service. The structure of the service system may be such that there is a single
service facility, or there are multiple service facilities in parallel, with a single queue
or with multiple queues. Multiple service facilities may be in series as well.
The speed with which service is provided may be expressed either as service rate,
which describes the number of customers services during a particular time period, or
as service time, indicating the amount of time needed to service a customer. The two
are reciprocal of each other. The speed of service may be fixed or variable.
3.
Queue structure: This refers to the order in which customers are picked up for
service. The queue discipline may be first-come-first-served (FIFO), last-come-firstserved (LIFO), service in random order (SIRO), priority service etc.
4. Customer behaviour: The customers may be patient by behaviour (who come, stand
in the queue, get the service and leave) or impatient, where they may be jockeying
between queues, for example.
Operating Characteristics of Queuing System
When analysing a queuing system, the commonly considered characteristics include:
(i) Average number of customers in the queue waiting for their service, called
queue length.
(ii) Average number of customers in the system: those in queues plus those getting
the service – the service length.
(iii) Average time a customer waits in a queue before getting service.
(iv) Average time a customer spends in the system.
Queuing Models
The queuing models may be categorised as being deterministic or probabilistic. If
customers arrive at known times and require service times that are known with certainty,
then the queuing system is deterministic and easy to analyse. Among the probabilistic
models, where element of uncertainty is present, the three models are:
1. Poisson-exponential single server model with infinite population: It is based on the
following assumptions:
(a)
(b)
(c)
(d)
(e)
The arrivals follow Poisson distribution, with a mean arrival rate of, say, λ.
The service time has exponential distribution. Let the service rate be µ.
Arrivals are from infinite population.
The customers are served on a first-come-first-served basis.
There is only a single service station.
Under these conditions, the various operating characteristics are as follows:
i.
Probability that the system is busy, ρ = λ / μ
ii.
Probability that the system is idle, P (0) = 1 – ρ
iii.
Probability that there are n customers in the system, P ( n ) = ρn (1 – ρ)
iv.
Probability that there are more than n customers in the system, P ( > n ) = ρn+ 1
v.
Expected number of customers in the system, Ls = ρ / (1 – ρ)
vi.
Expected number of customers in the queue, Lq = ρ2 / (1 – ρ)
vii.
Expected length of non-empty queues, Lq, = 1 / (1 – ρ)
viii.
Expected waiting time of a customer in the system, Ws = 1 / (μ – λ)
ix.
Expected waiting time of a customer in the queue, Wq = ρ / (μ – λ)
x.
Probability that a customer spends more than t units of time in the system,
Ws(t) = e- t/ws
xi.
Probability that a customer spends more than t units of time in the queue,
Wq(t) = ρ e- t/ws
The following points may be noted:
(a) For a system to be workable, it is necessary that λ < μ.
(b) The ratio ρ = λ / μ is called traffic intensity or the utilisation parameter. The
greater its value, the more frequently busies the system.
(c) The operating characteristics given above are applicable when the system is in
the steady state.
The information provided by a queuing model can be usefully employed in determining
the optimal level of service. At lower levels of service, the cost of providing service is
low but the waiting cost is high, while at higher levels of service, the service cost
becomes high but the cost of waiting is low. Thus, a balance may be struck and the
service level may be determined that would minimise the total cost.
2. Poisson-exponential single server model - Infinite population:
It is based on the
assumptions as the earlier model with the difference that the source population is small,
finite. The operating characteristics are summarised here:
With M as the number of customers in the source population, 1/ λ as the average interarrival time between successive arrivals and μ as the service rate,
(i) Probability that the system will be idle,
!1
(ii)
(iii)
(iv)
(v)
(vi)
i
#M
$
P(0) = ( ' %* M ! %* &+ &+ )
. i " 0 , ( M ! i )! , - - /
Probability that there are n customers in the system,
n
0
M!
%
&
01n2M
3 P (0) * +
P(n) = 4
, - ( M ! n )!
360
n5M
Expected length of the queue,
7
Lq = M !
(1 ! P ( 0))
Expected number of customers in the system,
Ls = Lq + (1 – P(0))
Expected waiting time of a customer in the queue,
Lq
Wq =
(1 ! P ( 0))
Expected time a customer spends in the system,
Ws = Wq + 1
3. Poisson-exponential, multiple server model—infinite population
The assumptions of this model are:
(a) The arrival of customers follows Poisson probability law, the average arrival rate
being .
(b) The service time has exponential distribution.
(c) There are K service stations each of which provides identical service.
(d) A single waiting line is formed.
(e) The input population is infinite.
(f) The service is on a first-come-first-served basis.
(g) The arrival rate is lower than the combined service rate of the K servers.
The operating characteristics are given here:
(i) Utilisation factor of the system, = K
(ii) Probability that the system is idle,
!1
( / ) K $
# K ! 1 ( / ) i
P(0) = ( '
7
K !(1 ! ) )/
. i " 0 i!
(iii) Probability that there are n customers in the system,
( / ) n
P(n) = P(0)
when n 2 K
,
n!
( / ) n
P(n) = P(0)
,
when n > K
K !K n ! K
(iv) Expected number of customers in the queue,
( / ) K P (0)
Lq =
K !(1 ! )2
(v) Expected number of customers in the system,
Ls = Lq + (vi) Expected waiting time in the queue,
Lq
Wq =
(vii) Expected time a customer spends in the system,
Ws = Wq + 1