Journal of Basic Physical Research ISSN; 2141-8403 PRINTS, 2141-8411 ONLINE Vol. 2, No. 1, pp 50-56, June, 2011 Available online at www.jbasicphyres-unizik.org
USING MULTI SERVER QUEUEING MODEL TO IMPROVE SERVICE DELIVERY *M.O. Onyesolu, I. E. Onyenwe and D. Edebeatu Department of Computer Science, Faculty of Physical Sciences, Nnamdi Azikiwe University, Awka (Accepted: 5th December, 2010)
ABSTRACT The endless customers waiting for service delivery is a phenomenon that bothers both the management and the customers alike. This study attempted to uncover the impact of increase in service rate or increase in the number of available server(s) on the performance of the system. The study reveals that though an increase in service rate improve service delivery, an increase in available number of servers improved service delivery better and reduced waiting times.
KEY WORDS: Queue, model, steady-state, performance, equilibrium. INTRODUCTION Waiting line is what we encounter everywhere we go, while shopping, checking into hotels, at hospitals and clinics, traffic lights, in banks’ tellers, fast- food joints, barber shops, bus stands, etc [1 and 2]. In a traditional non-queuing environment, customers can be left confused as to what line to stand in, what counter to go to when called and distracted by noisy crowded environment [2]. Therefore, we encounter queues in our everyday lives. In situations where facilities are limited and cannot satisfy the demand made upon them, bottlenecks occur which manifest as queue [1]. When customers wait in queue, there is the danger that waiting time will become excessive leading to the loss of some customers to competitors [3]. Queue results whenever people or items from different destinations or sources gather or converge at a point (or facility) in order to obtain related services. The rate at which people or items gather and the capacity of the service facility or facilities determine the size of the queue [4]. Studies related to queues have been carried out by researchers such [1, 5, 6 and 7]. Queueing theory has applications in diverse fields, including telecommunications, traffic engineering, computing and the design of factories, shops, offices and hospitals [8]. In queueing theory, a queueing model is used to approximate a real queueing situation or system so that the queueing behaviour can be analyzed mathematically. Queueing models allow a number of useful steady state performance measures to be determined. These performance measures include the average number on the queue or the system, the average time spent in the queue or the system, the statistical distribution of those numbers or times, the probability the queue is full, or empty, and the probability of finding the system in a particular state. These performance measures are important as issues or problems caused by queueing situations are often related to customer dissatisfaction with service. Sometimes it may be the root cause of economic losses in businesses. Analysis of the relevant queueing models allows the cause of queueing issues to be identified and the impact of proposed changes to be assessed. MATERIALS AND METHODS Waiting line at the Senior Staff Canteen, Nnamdi Azikiwe University, Awka, South Eastern Nigeria was studied over a period of 6 months. Senior Staff Canteen sells varieties of foods (rice, garri, semovita, with egusi soup, vegetable soup, and bitter leaf soup), and soft drinks, different brands of beer, as well as a limited number of specialty items (such as ukwa) on specified days. This canteen operated a single channel waiting line for a period of four months and a multi channel waiting line for another period of *Correspondence Author. Email:
[email protected]
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two months to determine the best method to be adopted to improve service delivery, reduce waiting time and optimize profit. The arrivals at the Senior Staff Canteen occur randomly and independently of other arrivals. This obeyed the Poisson probability distribution. The service time followed an exponential probability distribution, while the queue discipline is first come, first served. Probabilities and performance measures of the models were determined analytically as indicated in [5 and 8]. DETERMINATION OF PROBABILITIES AND PERFORMANCE MEASURES IN SINGLECHANNEL MODEL In this model, the following deductions are made when it is in a steady state [5 and 8]: Calculations are made as to determine the number of individuals (items) on a queue system, the number of individuals (items) being served, number of individuals (item) on the queue instead of estimating the number of individuals (items) on the queue system in probabilistic terms. a. Probability P of queueing on arrival is equal to the traffic intensity. Therefore, P(queueing on arrival) = ρ (1) where ρ is the utilization factor or traffic intensity. b. Probability Po that there will be no queue on arrival. Po = 1- ρ
(2)
This is the same as the probability of meeting nobody in the queue. c. Probability of there being n individuals in the system is given by Pn = (1- ρ) ρn
(3)
d. Probability of there being n or more individuals in the system is given by Pn = ρn (4) e. The expected (average) number of individuals (items) in the system is expressed as: (5) f. The expected (average) number of individuals (items) on the queue is expressed as: (6) g. The service time is given as: (7) Where µ is the mean service rate. h. The mean waiting time on the queue is the time an individual spends waiting for service. This is expressed as: (8) Where λ is the mean arrival rate and µ is the mean service rate. 51
Onyesolu, Onyenwe and Edebeatu
i. The average time in the system is the total waiting time in the system. It is the average time an individual (item) spends on the waiting line and the average time required for service. That is: (9) DETERMINATION OF PROBABILITIES AND PERFORMANCE MEASURES IN MULTICHANNEL MODEL In this model, the following deductions are made when it is in a steady state [5 and 8]: a. Probability Po that there will be no queue on arrival. (10)
where n is the number of individuals in the system, C is the number of service channels and ρ is the utilization factor or traffic intensity. b. Probability P of queueing on arrival is given as: (11) c. Probability of n individuals are in the system at a given point in time is given as: for n≤c
(12)
for n>c
(13)
d. Average length of queue or number of individuals on queue is given as: (14)
e. Average (or expected) number of individual in the system is expressed as: (15) f. Average waiting time on the queue is: (16) g. Average waiting time in the system is: (17) SINGLE-CHANNEL METHOD With this method, a server takes a customer's order, determines the total cost of the order, takes the money from the customer, and then fills the order. Once the first customer's order is filled, the server takes the order of the next customer waiting for service. MULTI-CHANNEL METHOD With this method, there are two servers. These two servers run in parallel. They take customers order as indicated in the single channel method. 52
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Although Senior Staff Canteen would like to serve each customer immediately, at times more customers arrive that can be handled by the Senior Staff Canteen food service staff. Thus, customers wait in line to place and receive their orders. Senior Staff Canteen is concerned that the methods currently used to serve customers are resulting in excessive waiting times. The management wants to conduct a waiting line study to help determine the best approach to reduce waiting times and improve service. The study conducted at the Senior Staff Canteen showed that the mean arrival rate is 45 customers per hour. For a one minute period, the mean arrival rate would be λ = 45 customers/60 minutes = 0.75 customers per minute. The probability of x customer arrivals during a one minute period can be computed using the Poisson probability function:
(18) where x = the number of arrivals in the time period λ = the mean number of arrivals per time period Using an exponential probability distribution, the probability that the service time will be less than or equal to a time of length t is: (19) where µ= the mean number of units that can be served per time period At the Senior Staff Canteen, the study showed that a single food server can process an average of 60 customer orders per hour. On a one minute basis, the mean service rate would be µ= 60 customers/60 minutes = 1 customer per minute. Therefore, the probabilities that an order can be processed in half a minute, one minute, and two minutes are: P(service time ≤ 0.5 min.) = 1-e-1(0.5)=1-0.6065=0.3935 P(service time ≤ 1.0 min.) = 1-e-1(1.0)=1-0.3679=0.6321 P(service time ≤ 2.0 min.) = 1-e-1(2.0)=1-0.1353=0.8647 STEADY-STATE OPERATION When the Senior Staff Canteen opens in the morning, no customers are in the restaurant. Gradually, activity builds up to a normal or steady state. The beginning or start-up period is referred to as the transient period. The transient period ends when the system reaches the normal or steady-state operation. The model adopted the steady-state operating characteristics. RESULTS AND DISCUSSIONS Table I showed the probability of x customer arrivals during a one minute period using the Poisson probability function and the result of the steady state probability of the same system. Table II showed the operating characteristics of Senior Staff Canteen for the two months when the service rate was 60 customers per hour and the next two months when the service rate increased to 75 customers per hour. The operating characteristics when an additional server was introduced into the system are shown in table III.
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Table I: Poisson probabilities for the number of customer arrivals at the Senior Staff Canteen during a one-minute period (λ = 0.75) and Steady-state probabilities of n customers in the Senior Staff Canteen (ρ = 0.75) Number of Arrivals 0 1 2 3 4 5 or more
Probability (Transient State) 0.4724 0.3543 0.1329 0.0332 0.0062 0.0010
Probability (Steady State) 0.2500 0.1875 0.1406 0.1055 0.0791 0.0593
The transient state of the model showed that the probability of no customers in one-minute period is 0.4724. The probability of one customer in one-minute period is 0.3543. The probability of two customers in one-minute period is 0.1329. When the system is at a steady state, there is a 0.25 probability that on arrival there would no customer waiting for service. Again, there is a 0.1875 probability that on arrival one customer would be seen to wait to be served. When the system is operating in a transient state, the likelihood that a customer will be served on arrival is high. This indicated that a customer will spend less time in the system and will be served quickly on arrival. On the other hand, when the system is operating in a steady state, the likelihood that a customer will be served on arrival is low. This indicated that a customer will spend more time in the system and will be served quickly on arrival Table II: Single channel operating characteristics with mean service rate µ = 1.00 customer per minute and mean service rate increased to µ = 1.25 customers per minute Operating Characteristics Probability of no customer in the system (P0) Probability of waiting on arrival (Pw) Number of customers in the system (L) Number of customers on queue (Lq) Time spent waiting in the system (W) Time spent waiting on queue (Wq)
Single Channel µ = 1.00 customer/minute 0.25 0.75 3.00 customers 2.25 customers 4 minutes 3 minutes
Single Channel µ = 1.25 customers/minute 0.400 0.600 1.500 customers 0.900 customers 2 minutes 1.2 minutes
At steady state when service rate was 60 customers per hour, the probability that a customer will have to wait to be served is 0.75. This indicated that a customer will wait for 75% of the total time before being served. Since a total of 4 minutes is spent in the system, a customer would wait for 3 minutes on the queue. When service rate was increased to 75 customers per hour, i.e. 1.25 customers per minute with arrival still at 45 customers per hour, the probability that the server will be free (idle) increased to 0.4. This means that the server will be 40% free of its time. Therefore, customers are served quicker. Increasing the service rate reduced drastically the time spent in the system and the time a customer would have to wait to be served.
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Table III: The operating characteristics of senior staff canteen with the two models Operating Characteristics
Probability of no customer in the system (P0) Probability of waiting on arrival (Pw) Number of customers in the system (L) Number of customers on queue (Lq) Time spent waiting in the system (W) Time spent waiting on queue (Wq) Time spent being serviced
Single Channel µ = 1.00 customers/minute
Two Channels µ = 1.00 customers/minute
Single Channel µ = 1.25 customers/minute
0.25
0.4545
0.400
0.75 3.00 customers 2.25 customers 4 minutes 3 minutes 60 seconds
0.2045 0.8727 customers 0.1227 customers 1.1636 minutes 0.1636 minute 60 seconds
0.600 1.500 customers 0.900 customers 2 minutes 1.2 minutes 48 seconds
The operating characteristics of the Senior Staff Canteen for the first two months, the next two months and the last two months is shown in table III. The system improved quite a lot when an additional server was introduced into the system. While the server was 25% free in single channel with service rate of 1 customer per minute, the same server is 40% free when the service rate was increased to 1.25 customers per minute. When an additional server was introduced into the system, the servers were 45.45% free of the total time. An additional server reduced the probability to 0.2045. This indicated that customers are served quicker in multi-server model which has the same parameters with a single channel system. CONCLUSION Decisions involving the design of waiting lines are based on a subjective evaluation of the operating characteristics of the waiting line. The cost of operating a waiting line system is determined based on the decision regarding the system design on a minimum operating cost, because in business cost is minimized and profit maximized. The study showed that an increase in the service rate in a single channel queue reduced waiting times, an additional server in the system improved service delivery and in addition reduced waiting times. Management at the Senior Staff Canteen, Nnamdi Azikiwe University, Awka, South Eastern Nigeria should adopt multi-channel waiting line with an increase in the service rate of µ = 1.25 customers/minute. This will improve the performance measures considerably and thus, will improve service delivery. REFERENCES 1. Olaniyi, T.A. (2004). An Appraisal of Cost of Queuing in Nigerian Banking Sector: A Case Study of First Bank of Nigeria PLC, Ilorin. J. Business and Social Sciences, 9(1 & 2), 139-145. 2. Onyesolu, M.O. and Okonkwo, O.R.M. (2006). A Comparative Study of Waiting-Line Situations (Queues) in Nnamdi Azikiwe University, Awka, South Eastern Nigeria. J. Sci. Engr. Tech. 13(1), 6481-6487. 3. Kotler, P. (2000). Marketing Management: The Millennium Edition (10th edition), New Delhi, Prentice-Hall of India, 784 pp. 4. Onyesolu, M.O., Eze, F.U. and Okonkwo, O.R.M. (2006). Comparative Analysis of Single-Path Communication System and Multi-Path Communication System Using Queueing Theory Approach. African J. Sci. 7(1), 1403-1408. 5. Kleinrock, L. (1975). Queueing Systems-Volume I: Theory, John Wiley and Sons Inc., Canada, 411 pp 6. Kleinrock, L. (1976). Queueing Systems-Volume II: Computer Applications, John Wiley and Sons Inc., New York, 537 pp.
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