A Study of Service in Restaurant by using Queuing Model

The restaurants want to avoid losing their customers due to a long wait on the line. This shows a need of a numerical model for the restaurant management to understand the situation better. This paper aims to show that queuing theory satisfies the model when tested with a real-case scenario. We obtained the data from a restaurant. We then derive the arrival rate, service rate, utilization rate, waiting time in queue and the probability of potential customers to balk based on the data using Little’s Theorem and M/M/1 queuing model. We conclude the paper by discussing the benefits of performing queuing analysis to a busy restaurant.


Introduction
There are several factors for a restaurant to attract the customers. The most important factors are taste, cleanliness, the restaurant layout and settings. These factors, when managed carefully, will be able to attract plenty of customers. However, there is also another factor that needs to be considered especially when the restaurant has already succeeded in attracting customers. This factor is the customers queuing time. The queuing theory is the study of queue or waiting lines. Some of the analysis that can be derived using queuing theory include the expected waiting time in the queue, the average time in the system, the expected queue length, the expected number of customers served at one time, the probability of balking customers, as well as the probability of the system to be in certain states, such as empty or full. The waiting lines are a common sight in restaurants especially during lunch and dinner time. Hence, queuing theory is suitable to be applied in a restaurant setting since it has an associated queue or waiting line where customers who cannot be served immediately have to queue (wait) for service. Many researchers have previously used queuing theory to model the restaurant operation [2], reduce cycle time in a busy fast food restaurant [3], as well as to increase throughput and efficiency [5]. In this chapter we use queuing theory to study the waiting lines in a restaurant. The restaurant provides 20 tables of 6 people. There are 8 to 9 waiters working at any one time. On a daily basis, it serves over 400 customers during weekdays and over 1000 customers during weekends. This paper seeks to illustrate the usefulness of applying queuing theory in a real case situation. In 1908, Copenhagen Telephone Company requested Agner K. Erlang to work on the holding times in a telephone switch. He identified that the number of telephone conversations and telephone holding time fit into Poisson distribution and exponentially distributed. This was the beginning of the study of queuing theory. In this section, we will discuss two common concepts in queuing theory. Little's theorem [7] 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: Where, λ is the average customer arrival rate and T is the average service time for a customer. Consider the example of a restaurant where the customer's arrival rate (λ) doubles but the customers still spend the same amount of time in the restaurant (T). These facts will double the number of customers in the restaurant (L). By the same logic, if the customer arrival rate (λ) remains the same but the customers service time doubles this will also double the total number of customers in the restaurant. This indicates that in order to control the three variables, managerial decisions are only required for any two of the three variables. Three fundamental relationships can be derived from Little's theorem [6]: T increases if L increases or λ decreases Rust [8] said that the Little's theorem can be useful in quantifying the maximum achievable operational improvements and also to estimate the performance change when the system is modified. The data were obtained from a restaurant through interview with the restaurant manager as well as data collections through observations at the restaurant. The daily number of visitors was obtained from the restaurant itself. The restaurant has been recording the data as part of its end of day routine. We also interviewed the restaurant manager to find out about the capacity of the restaurant, the number of waiters, as well as the number of chefs in the restaurant. Based on the interview with the restaurant manager, we concluded that the queuing model that best illustrate the operation of a restaurant is M/M/1. This means that the arrival and service time are exponentially distributed (Poisson process). The restaurant system consists of only one server. In our observation the restaurant has several waiters but in the actual waiting queue, they only have one chef to serve all of the customers. For the analysis of a restaurant M/M/1 queuing model, the following variables will be investigated [6]:  Wq : average waiting time in the queue. We also found out from observation and discussion with manager that each customer spends 55 minutes on average in the restaurant (W), the queue length is around 36 people (Lq) on average and the waiting time is around 15 minutes. It can be shown using (7) that the observed actual waiting time does not differ by much when compared to the theoretical waiting time as shown below.
Next, we will calculate the average number of people in the restaurant using (1).

L = 2.22 cpm x 55 minutes = 122.1 customers
Having calculated the average number of customers in the restaurant, we can also derive the utilization rate and the service rate using (4).
Hence  With the very high utilization rate of 0.991 during dinner time, the probability of zero customers in the restaurant is very small as can be derived using (2).
The generic formula that can be used to calculate the probability of having n customers in the restaurant is as follows: P n = (1-0.991) (0.991) n = (0.019) (0.991) n We assume that potential customers will start to balk when they see more than 10 people are already queuing for the restaurant. We also assume that the maximum queue length that a potential customer can tolerate is 40 people. As the capacity of the restaurant when fully occupied is 120 people, we can calculate the probability of 10 people in the queue as the probability when there are 130 people in the system (i.e. 120 in the restaurant and 10 or more queuing) as follows:

Volume 5
Probability of customers going away = P (more than 15 people in the queue) = P (more than 130 people in the restaurant)

Evaluation
The utilization is directly proportional with the mean number of customers. It means that the mean number of customers will increase as the utilization increases.  The utilization rate at the restaurant is very high at 0.991. This, however, is only the utilization rate during lunch and dinner time on Saturdays and Sundays. On weekday, the utilization rate is almost half of it. This is because the number of visitors on weekdays is only half of the number of visitors on weekends. In addition, the number of waiters remains the same regardless whether it is peak hours or off-peak hours.  In case the customers waiting time is lower or in other words we waited for less than 15 minutes, the number of customers that are able to be served per minute will increase. When the service rate is higher the utilization will be lower, which makes the probability of the customers going away decreases.

Benefits
 This research can help a restaurant to increase their QOS (Quality of Service), by anticipating if there are many customers in the queue.  The result of this work may become the reference to analyze the current system and improve the next system. Because the restaurant can now estimate of how many customers will wait in the queue and the number of customers that will go away each day.  By anticipating the huge number of customers coming and going in a day, the restaurant can set a target profit that should be achieved daily.  The formulas that were used during the completion of the research is applicable for future research and also could be use to develop more complex theories.  The formulas provide mechanism to model the restaurant queue that is simpler than the creation of simulation model in [9,4].

CONCLUSION
This research paper has discussed the application of queuing theory of a Restaurant.
Here we have focused on two particularly common decision variables. From the result we have obtained that the rate at which customers arrive in the queuing system is 2.22 customers per minute and the service rate is 2.24 customers per minute. The probability of buffer flow if there are 10 or more customers in the queue is 15 out of 100 potential customers. The probability of buffer overflow is the probability that customers will run away, because may be they are impatient to wait in the queue. This theory is also applicable for the restaurant if they want to calculate all the data daily. It can be concluded that the arrival rate will be lesser and the service rate will be greater if it is on weekdays since the average number of customers is less as compared to those on weekends. The constraints that were faced for the completion of this research were the inaccuracy of result since some of the data that we use was just based on assumption or approximation. We hope that this research can contribute to the betterment of a restaurant in terms of its way of dealing with customers. As our future works, we will develop a simulation model for the restaurant. By developing a simulation model we will be able to confirm the results of the analytical model that we develop in this paper. In addition, a simulation model allows us to add more complexity so that the model can mirror the actual operation of the restaurant more closely [1].