123601-Article Text-338083-1-10-20151009
123601-Article Text-338083-1-10-20151009
123601-Article Text-338083-1-10-20151009
Abstract
The formation of waiting lines is a common phenomenon that occurs whenever the current
demand for a service exceed the current capacity to provide that service. Decisions
regarding the amount of service to provide must be made frequently in industries even
though, it is often impossible to predict accurately when units will arrive to seek service
and how much time will be required to provide that service. This paper reports on a study
conducted on five petrol stations in Benin City, namely: Oando petrol station Akpakpava, AP
petrol station Ugbowo, Total petrol station Iselu, NNPC petrol station Benin-Auchi Road and
NNPC Mega filling station Benin-Sapele Road. The average arrival rate of customers per
hour for the five petrol filling stations were obtained as 95.4, 97.4, 98.5, 99.6 and 177.8
respectively while the average departure rate of customers per hour were obtained as 86.2,
89, 89.7, 91.7, and 171.6 respectively. The results show that queues exist in each of the five
petrol stations. It was also observed that the waiting time in the queue and service time at
the five petrol filling stations decrease with increase in the number of servers.
of staff [9, 19]. The application of queuing Poisson distribution pattern with arrival rate
theory may be of particular benefit in of “λ” customers per unit of time. It is also
pharmacies with high volume outpatient assumed that they are served on a first-come,
workloads and/or those that provide multiple first-served basis by any of the servers. The
points of service. By better understanding service time are distributed exponentially,
queuing theory, service managers can make with an average of “µ” customers per unit of
decisions that increase the satisfaction of all time. Queue parameters are found using
relevant groups: customers, employees and Little’s laws, which states that the long term
management [9]. A queueing model was also average number of customers in a stable
used by Gorunescu et al. [20] and system is equal to the long term average
Siddharthan et al. [21] to determine the main effective arrival rate multiplied by the
characteristics of the access of patients to average time a customer spend in the system.
hospital, such as mean bed occupancy and the The utilization factor is the proportion of the
probability that a demand for hospital care is system's resources that is used by the traffic
lost because all beds are occupied. which arrives at it. It should be strictly less
than one for the system to be functioning
2. Materials and Method effectively.
2.1. The queueing system Data were collected based on the arrival rate
The data collected were based on the arrival and departure rate of cars in the five petrol
rate and departure rate of cars in the five filling stations studied in Benin-City area of
petrol filling stations studied. This process is Edo State and they include: Oando Filling
also known as the birth and death process Station, Akpakpava with two servers and first
and the term birth refers to the arrival of a in first out (FIFO) queue discipline, AP Filling
new customer into the queuing system while Station, Ugbowo with three servers, and first
the death refers to the departure of a served in first out (FIFO) queue discipline. Others
customer from the petrol filling station. The are, Total Filling Station Uselu, with four
queues in the petrol filling stations were that servers, and first in first out (FIFO) queue
of multi-server types {(M/M/s):(FCFS)} discipline and NNPC Filling Station, Benin-
where customers arrive according to a Auchi Road with five servers, and first in first
Poisson process with infinite source. Where out (FIFO) queue discipline, while data were
“M” is the arrival process, “M” is the service also collected from NNPC Mega filling Station,
process and “s” is the number of servers with Benin-Sapele Road with eight service points,
first come first serve (FCFS) queue discipline. and first in first out (FIFO) queue discipline.
The queueing systems are multiple queueing The data collected from the five petrol filling
systems with identical servers in parallel and stations studied are presented in Table 1.
it is assumed that the arrivals follow a
Table 1. Arrival rates and service rates for the five petrol stations
Petrol Filling Station
Oando Station, AP Station, Total Station NNPC Station Benin- NNPC Mega
Akpakpava Ugbowo Uselu Auchi Road Benin-Sapele Road
Period λ µ λ µ λ µ λ µ λ µ
(Cars) (Cars) (Cars) (Cars) (Cars) (Cars) (Cars) (Cars) (Cars) (Cars)
8-9am 92 81 103 95 100 96 104 100 165 160
9-10am 94 83 95 88 96 89 98 92 170 162
10-11am 98 90 98 90 99 90 105 95 180 173
11-12pm 87 85 102 95 104 98 91 89 200 190
12-1pm 91 80 97 82 98 85 102 93 160 160
1-2pm 90 84 105 98 95 87 99 90 156 150
2-3pm 102 90 90 87 102 98 101 93 185 179
3-4pm 95 81 94 90 92 80 97 90 174 170
4-5pm 104 95 92 80 99 90 98 95 190 184
5-6pm 101 93 98 85 100 84 100 80 198 188
8-9am 94 83 95 88 96 89 98 92 170 162
9-10am 92 81 98 90 99 90 91 89 174 170
10-11am 98 90 97 82 102 98 101 93 185 179
Ls 10.0
Ws
0.103 hr = 6.18 min. or
97.4
Therefore, expected waiting time in the 2.7. Eight server’s case (M/M/8)
Lq 8.0 The queue system in NNPC Mega filling
W
queue,
q
97.4 = 0.082 hr. = 4.92min. station is an eight serve queue system of
or M/M/8 case with FIFO and the queue
Expected service time, µ = 6.18 – 4.92 = parameters are obtained as follows:
1.26min. Average arrival rate of customers, λ =
1.62 5334/30 = 177.8 /hour or 178 customers per
Utilization factor s = (3)(1.26) = 0.33 hour
Expected waiting time in system,
2.5. Four server’s case (M/M/4) Ls 10.0
Ws
The queue system in Total filling station is a 177.8 0.056 hr. = 3.36 min.
four serve queue system of M/M/4 case with Expected waiting time in queue,
Lq
FIFO and the queue parameters are obtained Wq
8.0
= 0.045 hr. = 2.7 min.
177.8
as follows:
Average arrival rate of customers, λ = Expected service time, µ = 3.36 – 2.7 =
2955/30 = 98.5/hour (99 customers per 0.66min
hours) 2.96
Utilization factor factor s = (8)(0.66) = 0.56
Expected waiting time in the system,
Ls 10.0
Ws 3. Results and Discussion
98.5 0.102 hr = 6.12 min. or
Expected waiting time in the queue, The computed results for expected waiting
L time in the system, the expected waiting time
8.0
Wq q in queue, the average arrival rate of
98.5 = 0.081hr or 4.86min or
customers in the system and the mean service
Expected service time, µ = 6.12 – 4.86 = rate are presented in Table 2. The utilization
1.26min factor was also computed for each of the
1.64
filling stations and the results showed that it
Utilization factor = s = 0.33
(4)(1.26)
was less than one for each of the stations. The
λ = 98.5/hour = 1.64min results also revealed that there was a rush of
Þ= λ /sµ = 1.64/4 x 1.26= 0.33min customers into the system during the early
hours of the day, when people are rushing to
2.6. Five server’s case (M/M/5) their various offices; during lunch-break; at
The queue system in NNPC filling station is a the end of the day’s activities when people
five serve queue system of M/M/5 case with are retiring to their various homes; during
FIFO and the queue parameters are obtained the weekends as well as during scarcity
as follows: periods. The queue situation on Fridays and
Average arrival rate of customers, λ = weekends in the patrol stations was always
2988/30 = 99.6 /hour or 100 customers per different from other weekdays. It was
hour observed that motorists formed long queues
Expected waiting time in system, at the various stations waiting to be served.
Ls 10.0 Also the arrival rate was always found to be
Ws
99.6 0.10 hr. = 6 min. higher than the service rate for each of the
stations during the weekends
The results revealed that the average arrival society. The queue might involve data waiting
rate of customers to the filling stations was for processing, equipment parts waiting in an
always higher than the average departure assembly line or people waiting in line at
rate of customers from the filling stations and various types of business centers. Queue
this explained while queues always existed in theory is an important tool used to model
all the filling stations. This is clearly seen in many supply chain problems and it is used to
the graph of arrival rate and service rate for study situations in which customers form a
Oando and NNPC petrol stations in Figure 1 line and wait to be served by service or
which shows that the arrival rate was always manufacturing facility. The results from the
higher than the service rate. Figure 2 also study revealed that utilization factor for each
shows that the rate at which customers of the filling stations was less than one (1)
entered any of the stations for service was which showed that the servers were duly
always higher than the rate at which attending to their customers. The utilization
customers leave the station after being factors were obtained as 0.63, 0.33, 0.33, 0.28
served. This is the reason while there was and 0.56 for Oando filling station, AP Filling
always queue in any of the stations studied. Station, Total filling station, NNPC filling
However, the NNPC Mega filling station with station and NNPC Mega filling Station. It was
eight (8) service points has almost the same observed that only the NNPC Mega filling
arrival rate and service rate of customers. station was using all its service points. It was
This is closely followed by the therefore suggested to the filling stations to
NNPC filling station with five (5) service always endeavour to use all their service
points. points (pumps) at peak periods to reduce
queues at such stations
4. Conclusion
Queueing systems are more prevalent
in our increasingly congested and urbanised
Station)
Service Rate (Oando
Station)
Arrival Rate (NNPC Mega
Station)
Service Rate (NNPC Mega
Station)
Periods
Fig.1. Arrival rate and service rate for two petrol stations
Petrol Station
Fig. 2: Average arrival and departure rate for the five petrol stations
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