This work is supported with little’s theorem. Little’s theorem describes the
relationship between throughput rate (i.e arrival and service rate), cycle time
and work in process (i.e number of customers/jobs in the system). This relationship
has been shown to be valid for a wide class of queuing models. The theorem
states that the expected number of customers (N) for a system in steady state
can be determined using the following equation:
L = λT
Here, λ is the average customer
arrival rate and T is the average service time for a customer.
Three fundamental relationships can
be derived from Little’s theorem:
·
L increase if λ
or T increases
·
λ increases if L
increases or T decreases
·
T increases if L
increases or λ decreases
Little’s law can be applied in any system in which the
mean waiting time, mean line length (or inventory size), and mean throughput
(outflow) remain constant. To some extent this is an arbitrary decision, but in
most real-world situations, measuring the outflow of a queue is easier than
measuring its inflow.
Conceptual
Framework
A queuing
system is composed of the following components or parts each of which is
described below;
1.
Calling
population (or input source)
2.
Queuing process
3.
Queue discipline
4.
Service process
(or mechanism)
In
most cases, queuing models can be characterized by the following factors:
·
Arrival time distribution. Inter-arrival times most commonly fall into one of
the following distribution patterns: a Poisson distribution, a deterministic
distribution, or a General distribution and are most often assumed to be
independent.
·
Service time distribution: The service
time distribution can be constant, exponential, hyper-exponential,
hypo-exponential or general. The service time is independent of the inter-arrival
time.
·
Number of servers: The queuing calculations change depends on whether there is a single
server or multiple servers for the queue.
·
Queue lengths (optional): The queue in a system can be modeled as having
infinite or finite queue length.
·
System capacity (optional): The maximum number of customers in a system can be
from I up to infinity. This includes the customers waiting in the queue.
·
Queuing discipline (optional): There are several possibilities in terms of the
sequence of customers to be served such as FIFO, random order, LIFO or
priorities.
Data
were obtained from Next-time through interview with the restaurant manager as
well as observations at the restaurant.
Based on the interview with the supermarket manager,
we concluded that the queuing model that best illustrate the operation of Next-time
is M/M/I.
For
the analysis of the Next-time M/M/I queuing model, the following variables will
be investigated:
* λ: The
mean customers arrival rate
* μ: The
mean service rate
* p: λ/μ:
utilization factor
* Probability of zero customers in the
restaurant:
Po = 1
– P (2)
·
Pn: The probability of having n customers in the supermarket.
Pn = (1 – p)pn
·
L: average number
of customers shopping in the supermarket.
L = p = λ
1-p μ-λ
·
Lq: average
number in the queue.
Lq = L x p = p2 = pλ (5)
1 – p
μ-λ
·
We: average time
spent in Next-time, including the waiting time.
W = L = 1 (6)
λ
μ-λ
·
Wq: average
waiting time in the queue.
Wq =
Lq = p (7)
λ μ-λ