1 Introduction
Queuing research is very significant for improving service quality. Queuing involves medical treatment, call center, computer network, bank, traffic and other fields. When the capacity of the service system doesn't meet the service demand of customers, the service provider should increase the capacity. However, if the cost of increasing the capacity is very high, the queuing becomes inevitable
[1]. In order to improve queuing efficiency, customer satisfaction and reduce the total waiting cost in the service system, we need to make an effective distinction between the heterogeneity of customers (i.e., the waiting costs of different customers are different). Based on the research of
rules, it pointed out that from the perspective of reducing waiting cost in service system, customers with high waiting cost should be given priority, and corresponding levels of priority should be set according to the different waiting costs
[2-4].
The essence of classification service is the resetting of waiting time for customers. We can reduce the waiting time of the priority customer (VIP) by increasing the waiting time of the regular customer. However, it may decrease the satisfaction of the regular customer. Due to the fairness preference of people, regular customers may have unfairness feeling when they had perceived some customers reducing the waiting time by paying an extra service fee. This unfairness feeling may cause changes in behaviors of customers in queuing such as switching among different queues or choosing to leave, these behaviors are bound to affect the revenue and customer utility
[5, 6]. Existed researches
[7, 8] mainly considered the following problems on the basis of classifying customers with priority options in service system: 1) Should service providers adopt the mechanism of customer classification? How to classify customers? 2) Should the priority service be charged? How much do we charge?
The traditional research on the queuing system mainly focused on the mathematical calculation and evaluation of waiting time
[9]. Recently, people's preference behaviors were considered in queuing system, and some representative research had been formed
[10].
Existed researches
[11-13] showed that the fairness preference of customers came from the perception of transaction process (or the visualization of service process) and the psychological changes by the comparison. The description of fairness focused on two aspects: 1) Unfairness of expenditure (difference in cost of purchasing services or products); 2) Unfairness of acquisition (difference in utility or profit achieved from the purchasing services or products). The principle of fairness was mainly reflected in three aspects: 1) Whether the above two are the same; 2) The relationship between input and output; 3) Setting a certain psychological benchmark. Nazerzadeh and Randhawa
[14] presented that based on the heterogeneity of customer waiting cost, the service system could achieve asymptotic optimum by using the reasonable pricing to provide customers with two levels of service in the M/M/1 queuing system. In other words, whether the service system divides the customers into two categories or
categories, the revenue is almost same when the number of customers tend to infinity. Therefore, this paper only considers two categories of customers in the research process, and considers the psychological fairness produced by the comparison between the two queues.
In the existed researches, the criteria of fairness to the queuing mechanism mainly focused on two aspects: 1) First Come First Served (FCFS), this was the most widely recognized code of queuing in the world
[5, 6, 11]; 2) Shortest Job First (SJB), the purpose of this queuing mechanism was to minimize the system average waiting time
[15]. However, it is very difficult to measure the service time required by each customer in the actual operation. Furthermore, customers pay more attention to their own waiting time rather than the average waiting time in the service system. Based on the above principles, it also further illustrates that the classification service goes against the principle of FCFS, and inevitably brings about the unfairness feeling of customers. Existed researches about fairness in queuing system mainly focused on the unfairness between servers caused by the different levels of busyness among different servers
[16, 17].
Rafaeli, et al.
[6] and Alexander, et al.
[18] proposed that even if the customer paid for the priority with additional charge, the customer who did not purchase the priority still had unfairness feeling through empirical analysis. The existed researches about the behaviors of customers in queuing system mainly focused on the impatience of those customers who have high waiting time cost (priority customers)
[12, 19-21]. But the existed papers do not pay enough attention to the mentality of regular customers who are in a more "vulnerable" position in the queuing system. Because there are few researches about the customer fairness preference in queuing process, and existed researches also rarely consider the behavioral changes caused by the customer fairness preference in queuing system and the impact of these behavioral changes on revenue, customer utility and social welfare, this research first introduces the customer fairness preference into the queuing system and systematically analyzes the above problems.
The customer classification also brings the difference of service value. Afeche
[22] considered a single-queue system that customers were sensitive to the delay of service and service values of different customers were different, in which the service value of customer followed a continuous distribution. Afeche and Pavlin
[17] assumed that due to immediate delivery and delay cost, service values of customers were different in different types of queuing systems, and the customer service value followed a uniform distribution related to waiting cost. Based on the above researches, this paper reflects the heterogeneity of service value through comparing the difference of service value between the priority customers and the regular customers.
The innovation of this article: Building a new customers' utility model that takes into account the fairness preference of customers. By studying the impact of the fairness preference of regular customers in non-preemptive priority M/M/1 monopoly service system (weak product substitutability or weak market competition), we put forward the optimal service strategy (queuing mechanism scheme) and the corresponding service pricing based on different optimization objectives. This paper sets the fairness preference of regular customers as a parameter to reflect the customer's psychological changes. In some situations, the service provider sets up the VIP server and the regular customer server where they can be seen each other (such as boarding gate, bank, etc.), but sometimes sets up the VIP server and the regular customer server where they cannot be seen each other (such as network call center, etc.). Because customer fairness preference derives from the psychological effects of visualization or perception in the service process, this paper finds that the setting of the two queues (i.e., whether priority and regular customers can be seen each other) will affect the revenue and social welfare.
This research shows that: First, from the perspective of revenue maximization, the service provider should adopt the mechanism of customer classification and set up the two kinds of customers where they can be seen each other (observable queues). In this situation, as customer fairness preference increasing, more customers are willing to get the priority service by paying higher priority fee, so we should utilize the incentive of customer fairness preference to get more revenue. Second, from the perspective of customer utility maximization, the service provider should cancel the mechanism of customer classification, and adopt the strategy that keep one queue (regular customers) only. Finally, from the perspective of social welfare maximization, if the governor regards the revenue and the customer utility are same important, the service provider should adopt the mechanism of customer classification but set up the two kinds of customers where they cannot feel each other. In this situation, as the customer fairness preference increasing, the social welfare will gradually decrease. So the incentive of customer fairness preference should be minimized.
The main architectures of this article are as follows: First, a new utility model is constructed under the condition of fairness preference for regular customers. And the classification service pricing, revenue, total customer utility and social welfare functions are constructed. Then, from the perspectives of revenue maximization, total customer utility maximization and social welfare maximization, the corresponding objective functions are optimized and solved, and we analyze the features of the optimal solutions. Next, we compare the relationship among the above three optimal solutions and propose the corresponding management insights. Finally, we summarize the main results in this paper and present the further research.
2 The Model Setup
In this paper, we suppose that the arrival rate of the customer in the service system is (i.e., Poisson), the service rate is , the fee of each customer who enters the service system is , if the customer pays an extra fee with , he can join a special queue with priority. That is to say, customers with a total payout of have a priority over those paying only a entrance fee (this article assumes that the service process cannot be preempted, that is, priority customers can not interrupt the regular customers who have being served). We assume that the service value of priority customers is , the service value of regular customers is . In this paper, we only consider (i.e., the service value of priority customers is more than the regular customers).
Rafaeli, et al.
[6] and Alexander, et al.
[18] pointed out that in the queuing system, even if the customer paid for the priority with additional fee, the customer who did not purchase the priority still had unfairness feeling. Based on the existed social studies of the fair queuing mechanism (FCFS), we know that the queuing methods that violated the FCFS mechanism may lead to the unfairness feeling of customers. This paper assumes that customers need to decide which tickets to buy (which queue to join) before entering the service system (i.e., before seeing the queues). In this case, the customer will make a decision based on historical data, friends' recommendation and network evaluation. And his information is based on the average waiting time of the service system which can be obtained by accessing to the historical data, so this paper uses the average waiting time of different queues in the system to measure relevant effects. This paper uses
to express customer fairness preference, obviously,
; and uses
to express the difference of the average waiting time between the different queues. The most obvious contrast between customers in different queues is the difference of the waiting time. Based on the above principle, this paper uses
to express the negative utility of regular customer unfairness feeling. Because VIP customers in the service system have less waiting time, they have no psychological negative utility. This paper only considers the fairness preference of regular customers, and constructs the corresponding utility model based on it.
This paper assumes
is waiting cost per unit time. Based on the existed research
[17], this paper assumes that the waiting cost per unit time follows a uniform distribution in the interval
(we assume
and
by standardizing
). We assume that the waiting time of the customer who pay only a fee of
is
, and the waiting time with the customer who pay a fee of
is
. The priority customer utility
and the regular customer utility
are as follows:
What is different from the existed research is the regular customer utility , this is because this paper considers the negative utility caused by fairness preference of regular customers, which is also the innovation of this paper.
In the classification service system, the customer fairness preference will bring about the psychological negative utility, and the negative utility of the customer also affects the customer's behavior in the process of queuing. Therefore, we incorporate the psychological utility of the customer into the original customer utility model for constructing a new customer utility model.
In monopoly system, customers only have two options after entering the system: The priority queue or the regular queue. If customers choose to join the priority queue for service, must be hold. Therefore, the waiting cost per unit time for the customer who joins the priority queue satisfy . We assume the proportion of priority customers in the service system is . According to Eq.(1), we can get .
Next, we will consider the corresponding average waiting time for different queues. In the monopoly service system (customers can not choose to leave), according to the literatures [
23-
25], we can see that the average waiting time of priority customers and regular customers are as follows:
where is the service intensity (this paper only considers , that is, , otherwise the length of the queue is infinite).
3 Revenue Maximization
Based on the assumption that the waiting cost for customers follows a uniform distribution in the interval , the functional relationship among the extra payment fee for the customer to join the priority queue, the proportion of the priority customers and the customer fairness preference parameter is as shown in Equation (3):
We assume is the revenue function when the proportion of priority customers is and the customer fairness preference parameter is . And its expression is as shown in Equation (4):
To optimize the revenue function in , the corresponding response function is shown:
Then, we can get the following conclusions:
Lemma 1 1) If , there are two types of customers in the system when the company gains the maximum revenue. At this point, the cost of joining the priority queue the optimal pricing of service provider and the maximum revenue are as follows:
For the above optimal solutions, we can get:
Therefore, the optimal proportion of priority customers, the priority fee, and the maximum revenue all increase as the customer fairness preference increases.
If , all customers join the priority queue, that is, there are only priority customers in the system. At this point, the priority fee and the maximum revenue are as follows:
Obviously, the cost of joining the priority queue and the maximum revenue increase as the customer fairness preference increases.
If existed research, the optimal proportion of priority customers, the optimal pricing of classification service and the maximum revenue in the service system are shown as Equation :
In a monopoly service system, customers must choose one from the two queues, so the service provider can increase revenue by increasing unlimitedly. When the service provider cancels the mechanism of customer classification i.e., and the regular customer is only reserved only one kind of customer and the queuing mechanism is FCFS, the maximum revenue is .
We can get two special situations as follows:
1) All customers join the priority queue: The service provider adopts the mechanism of customer classification by charging a certain additional fee. Because the customer fairness preference is large, all customers choose to join the priority queue by paying the priority service fee. There is no customer in the regular queue.
2) Only keep regular customers: The service provider does not classify customers and does not charge any classification service fee.
To sum up the above situations, we can get the following relation:
Then we can get the following conclusions and management insights.
Proposition 1 In the monopoly service system, even if the service provider can charge a very high value of to increase revenue, from the perspective of revenue maximization, the service provider should still adopt the mechanism of customer classification and set up the two queues where they can see each other. We should incentive and enhance customer fairness preference by direct comparison between the two queues to gain the maximum revenue.
Because the customers can only choose to join the regular queue or priority queue in the monopoly service system, the regular customers can only choose to transfer into the priority queue when the unfairness feeling causes a strong negative utility, then more and more customers join the priority queue. The service provider can also utilize the customer psychology to charge higher priority service fee for increasing revenue.
Based on the new utility model, this paper finds some important results and management insights from the perspective of revenue maximization. Next, we will take an analysis about the customer utility maximization.
4 Customer Utility Maximization
According to the analysis of customer utility in Section 2, if we adopt the mechanism of customer classification, the total customer utility function in service system is as follows:
Through the Equation (10), we can get the total customer utility function in the service system. By optimizing the in , we get the response optimization function as shown:
Therefore, obtains its maximum value at the boundary of its definition domain (i.e. or ).
Lemma 2 If , there are only regular customers in service system, that is , we can get
If , there are only priority customers in service system, we can get
It is clear that the two above situations satisfy the following relationship: .
Meanwhile, there is . Therefore, when there is only regular customer in service system, the customer can get the maximum utility.
For the monopoly service system, we can get the following conclusions and management insights.
Proposition 2 From the perspective of maximizing the total customer utility, the service provider should cancel the mechanism of customer classification and only retain regular customers. At this situation, the service rule provided by the service system is converted into First Come First Served FCFS.
In the monopoly service system, whether the service provider adopts the mechanism of customer classification or not, the average waiting time of the whole system is unchanged. But when the service provider adopts the mechanism of classification service, he will charge an additional fee, which causes a reduction in the total utility of customers in service system. At the same time, with the increase of customer fairness preference, the total utility of customers will decrease.
From the above analysis, we know that the revenue increases as the customer fairness preference increases but the customer utility decreases as the customer fairness preference increases, what's more, the service mechanisms of the above two are also different. So we will consider the combination of the above two (i.e., social welfare) in the following part.
5 The Social Welfare Maximization
The social welfare is derived from revenue and customer utility in service system. Therefore, we can get the social welfare equation in the service system as follows:
5.1 Optimal Solution and Property Analysis
We optimize the social welfare in , and get the following conclusions:
Lemma 3 If , the optimal pricing of classification service and the maximum social welfare are as follows:
For the above optimal solutions, we can get:
Obviously, both the optimal proportion of priority customers and the optimal service pricing increase as the customer fairness preference increases in service system. But the social welfare in the service system decreases as the customer fairness preference increases. Therefore, from the perspective of maximizing social welfare, the service provider should minimize the negative effect of customer fairness preference when he adopts the mechanism of classification service.
If , the optimal pricing of classification service and the maximum social welfare are as follows:
Obviously, as the customer fairness preference increases, the maximum social welfare is fixed when all customers in the service system join the priority queue. In this situation, that is to say, the maximum social welfare does not change with customer fairness preference although the cost of joining the priority queue can still increase as the customer fairness preference increases.
If , the optimal proportion of priority customers, the optimal pricing of classification service and the maximum social welfare in the service system are shown as Equation :
When the service provider cancels the mechanism of customer classification i.e., , the regular customer is only reserved only one kind of customer and the queuing mechanism is FCFS, i.e., . At this time, the maximum social welfare in the service system is: .
Obviously, when there is only one queue in the system, we can get .
To sum up the above situations, we can get the following relation:
therefore, the service provider should adopt the mechanism of customer segmentation. At the same time, we also find that whether all customers join the priority queue or the system has only regular customers, the maximum social welfare is same.
So we can get the following conclusions and management insights.
Proposition 3 From the perspective of maximizing social welfare, the service provider should adopt the mechanism of customer classification. At the same time, the priority queue and the regular queue should be set up where they cannot be seen each other Unobservable queues. So we can reduce the cost of joining the priority queue by reducing customer fairness preference for maximizing social welfare.
The social welfare is composed of the revenue and the total customer utility. From above conclusions, we know that the optimal proportion of priority customers, the priority fee, and the maximum revenue all increase as the customer fairness preference increases. Therefore, as the customer fairness preference increases, more and more customers join the priority queue, and the waiting time of all customers regular customers and priority customers also increases correspondingly. The increasing priority fee and waiting time will cause a more reduction about the total customer utility. This paper finds that, as the customer fairness preference increases, the rate of reduction about the total customer utility is bigger than the rate of increase about the revenue. So the social welfare will decrease as the customer fairness preference increases.
5.2 Further Study
In Subsection 5.1, we consider the social welfare equation in the service system as follows: . It means that the revenue and the customer utility are same important. But in some situations, the governor may think that the revenue and the customer utility are not same important. So we can assume the new social welfare equation in the service system as follows: . That is to say, in the aspect of social welfare, some governors will regard the revenue is more important than the customer utility (i.e., ) but others maybe regard the revenue is less important than the customer utility (i.e., ).
From the optimal solution and property analysis in section 5.1, we know that the social welfare will decrease as the customer fairness preference increases when . But this paper finds that if , the social welfare will increase as the customer fairness preference increases.
We assume . Therefore, from the perspective of maximizing social welfare, if the governor thinks that the revenue and the customer utility are not same important and , the service provider should not only adopt the mechanism of customer classification but also the priority queue and the regular queue should be set up where they can be seen each other (observable queues).
6 Conclusions and Further Work
In monopoly service system, this paper considers the customer fairness preference caused by customer classification based on a stylized non-preemptive M/M/1 queue. This paper conducts the optimization analysis from three different perspectives: Revenue, total customer utility and social welfare. And we get some valuable conclusions and management insights.
The mainly work of this paper:
From the perspective of revenue maximization, the service provider should adopt the mechanism of customer classification (i.e., customers are divided into two kinds: Priority customer and regular customer). And the service provider should set up the two kinds of customers where they can be seen each other (observable queues).
From the perspective of total customer utility maximization, the service provider should cancel the mechanism of customer classification, and adopt the strategy that keep only one type of queue (i.e., regular customers).
From the perspective of social welfare maximization, the service provider should also adopt the mechanism of customer classification. But if the revenue and the customer utility are same important (i.e., ), the two kinds of customers must be set up where they cannot be seen each other (unobservable queues); if the revenue and the customer utility are not same important and , the two kinds of customers must be set up where they can be seen each other (observable queues).
The above conclusions are obtained based on both different customer waiting costs and different customer service values. Obviously, the conclusions of this paper are applicable to the situation that the service values of different kinds of customers (priority customers and regular customers) are same.
The main work of this paper is summarized in Table 1.
Table 1 The optimal service strategy and classification service pricing of three different optimization objectives |
optimization goal | revenue | customer utility | social welfare () |
service strategy | two queues (observable queues) | one queue (regular queue) | two queues (unobservable queues) |
pricing of priority | | | |
This paper considers the service system is monopoly system (i.e., customers must choose one queue to join), but customers may have another choice (i.e., leave the current service system) in real life. At the same time, in this study, we do not consider the impact of the fairness preference of priority customers on revenue, customer utility and social welfare, which are our further research work.
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