We consider the equilibrium behavior of customers in the
Due to wide applications for management in service system and modern electronic commerce, there exists an emerging trend to study the behavior of customers in queueing systems. In these models, the customers decide to join or balk in a decentralized way. Their decisions are affected by how much system information can be observed and how other customers make their decisions. This can be viewed as a game among the customers. Researches about the economic analysis of queueing models can go back at least to the pioneering work of Naor [
The concept of an
Guo and Hassin [
Queues with setup times have also been investigated by many scholars [
Frequent setups will inevitably lead to the operating cost being too high, so it is very crucial to determine when the server should start his service in many practical queueing systems. By utilizing
The structure of this paper is organized as follows. In Section
We assume that there is a single server queue with infinite capacity where customers arrive according to a Poisson process with rate
We denote the state of the system at time
Transition rate diagram of the original model.
In this paper, we suppose that every customer gets a reward of
We now commit to the fully observable scenario, which assumes that customers observe not only the server state, but also the exact number of customers in the system. We will study the equilibrium threshold strategies and stationary probabilities for the fully observable case as well as some interesting performance measures of this system.
In order to study the general threshold strategy adopted by all customers in the fully observable case, we will consider the mean overall sojourn time of the customer who is confronted with different server state upon arrival firstly. We start with the fully observable case where customers know the exact state of the system and a customer in state
(I) If a new customer finds the system in state
(II) If a new customer finds the system in state
(III) If a new customer finds the system in state
Because any customer in
In the fully observable
From (
We now turn our attention to the stationary distribution probabilities. Be aware that if all customers follow the threshold strategy in Theorem
Transition rate diagram for the
The corresponding stationary distribution
In the fully observable
The stationary distribution of the system is gained from the following balance equations:
From (
By putting
Theorem
We now study an expected cost function per unit time for the
The busy cycle for the fully observable case is denoted by
By using the definitions of all costs listed above, the expected cost function per unit time per customer is given by
We turn our attention to the fully unobservable case, where the customers do not observe the state and system size at all when they join the system. We will study the equilibrium arrival rate and stationary probabilities for the fully unobservable case as well as some interesting performance measures of this system.
We will prove that there exists a mixed equilibrium strategy. An arriving customer joins the fully unobservable system with a certain probability
Let
In the fully unobservable
The balance equations are presented below:
From (
In the fully unobservable
Let
Let
Next, according to the value of
When
In the fully unobservable
This conclusion is consistent with Theorem
When
It is clear that when
(a) If
According to Proposition
Numerical examples for
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0.0211146 | 0.0418168 | 0.0626556 | 0.0837833 | 0.105319 | 0.127408 | 0.150264 | 0.174238 | 0.2 |
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0.378885 | 0.377214 | 0.375112 | 0.372582 | 0.369567 | 0.365949 | 0.361532 | 0.355973 | 0.34861 | |
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0.38 | 0.378885 | 0.377214 | 0.375112 | 0.372582 | 0.369567 | 0.365949 | 0.361532 | 0.355973 | 0.34861 |
Numerical examples for
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0.0206732 | 0.0408608 | 0.0610673 | 0.0814019 | 0.101923 | 0.122683 | 0.14374 | 0.165172 | 0.187081 |
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0.479226 | 0.478046 | 0.476574 | 0.474841 | 0.472846 | 0.470565 | 0.467952 | 0.464943 | 0.461439 | |
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0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 |
Numerical examples for
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0.0391792 | 0.0583989 | 0.0583989 | 0.0776041 | 0.096822 | 0.116065 | 0.135342 | 0.154658 | 0.174019 |
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0.95934 | 0.958828 | 0.958828 | 0.958219 | 0.957533 | 0.95678 | 0.955963 | 0.955084 | 0.954143 | |
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0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 |
Our objective is to give some interesting performance measures of the fully unobservable system and determine the optimum value of the control threshold
In this section, we obtain some numerical experiments to show the different effects of the fully observable and unobservable information systems. We let
System size
Social benefit
Cost function
In Figure
In Figure
In Figure
In this paper, we studied the equilibrium behavior of customers in
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is partially supported by CPSF (2015M572327), NSFC (71131003, 71371075, and 71271089), and NSFA (KJ2013B162, KJ2013A194, and KJ2014A174). It is also supported by Chaohu University Scientific Research Fund. The authors would like to sincerely thank the anonymous referees and editors for their valuable comments and suggestions which are very helpful for them to improve the presentation of the paper.