Should primary user be given preemptive priority in cognitive radio networks?

Accepted Manuscript Should primary user be given preemptive priority in cognitive radio networks? Sheng Zhu, Jinting Wang

PII: DOI: Reference:

S0140-3664(18)30052-5 https://doi.org/10.1016/j.comcom.2018.10.002 COMCOM 5782

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Computer Communications

Received date : 18 January 2018 Revised date : 5 October 2018 Accepted date : 9 October 2018 Please cite this article as: S. Zhu, J. Wang, Should primary user be given preemptive priority in cognitive radio networks?, Computer Communications (2018), https://doi.org/10.1016/j.comcom.2018.10.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Should primary user be given preemptive priority in cognitive radio networks? I Sheng Zhua,b , Jinting Wanga,∗ b School

a Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454003, China

Abstract Cognitive radio (CR) networks with a single primary user (PU) and multiple secondary users (SUs) are often modeled as a priority queueing system, in which PU has higher priority over SUs. It is an interesting problem whether or not PU should preempt the SU in service. We study the optimal policy from two different perspectives, namely, throughput and social welfare of the system, based on users’ strategic behavior. We first study users’ equilibrium strategic behavior and derive two dimensional equilibrium joining strategies under a natural cost-reward structure, and then the throughput and the social welfare in the preemptive mechanism are compared with those corresponding to nonpreemptive case. It is surprising to find out that the nonpreemptive case is better than the preemptive case in some situations from the viewpoint of throughput maximization, but in other situations permitting preemptions is an optimal decision. While from the perspective of the social welfare, we observe that the nonpreemptive case is always better than the preemptive case. These results provide important managerial insights into how to design the service mechanism and control SUs in the CR systems. Keywords: Cognitive radio networks, Priority queue, Nash equilibrium, Throughput, Social welfare

1. Introduction Due to the rapid increase of spectrum demand, spectrum has become a scarce resource for the wireless communication [1]. Under the conventional static spectrum access scheme, the utility of spectrum is extremely low [2]. To solve spectrum scarcity, Federal Communications Commission (FCC) recommended to improve spectrum usage by efficiently allocating frequency channels to unlicensed users (i.e., secondary users (SUs)) without impacting licensed users (i.e., primary users (PUs)) [3]. Based on this proposal, the so-called cognitive radio (CR) technique was developed. As a dynamic spectrum access (DSA) strategy [4], the CR technique improves the spectrum usage and release idle spectrum resources for SUs. It has been proved to be a prominent technique to solve the low utilization issue of I This ∗

work was supported by the National Science Foundation of China (Grant nos. 71571014 and 71871008). Corresponding author. Fax: +86-10-51840433. Email address: [email protected] (Jinting Wang)

Preprint submitted to Elsevier

October 6, 2018

spectrums. In a CRN, SUs can opportunistically access the spectrum resource owned by PUs according to the DSA protocol [5–7]. Obviously, the spectrum occupancy policy plays a key role in analyzing the system performance of CRNs [8]. How to model the process of spectrum access? It is an interesting issue and has been widely studied [9–11]. In the last decade, the use of queueing system to model CRNs has become a new trend. Various queueing models were used to model CRNs in different situations. For example, the CRNs were modeled as a priority queueing system in [9] and as an M/G/1/K queueing model in [10]. [11] proposed the unreliable queueing system characterizing the CRNs with imperfect spectrum sensing. In addition, [12] assumed that PUs leave the system immediately if the corresponding spectrums are not available, and then modeled the spectrum access of SUs as the retial queueing system. Whereafter, [13] and [14] considered a more general case, i.e., there exist a PU buffer and an SU buffer in the system. In this situation, a preemptive priority queueing system with two infinite buffers was adopted. Based on the queueing theory, a large amount of literatures focused on the network stability of CRNs [15], performance analysis [9, 10], throughput [14, 16], quality of service [17, 18] and optimal pricing [5], etc. Recently, some works focused on the management of CRNs from the perspective of social welfare maximization [19]. In a CRN, users’ strategic behavior has significant impact on the system performance and there has been an emerging trend to study CR systems related to economic analysis and strategic customer behavior. In the process of analyzing the strategic behaviors of PUs and SUs, the so-called game approach was often adopted. [20] considered the game between a spectrum service provider and its end users in CRNs and obtained the subgame perfect nash equilibrium. [21] studied the service selection of users by using Nash equilibrium. [13] believed the game between PUs and SUs is the Stackelberg game and the game among PUs (or SUs) is the noncooperative game, and then got users’ strategic behavior in CRNs with different information levels. Interested reader can refer to [1, 11, 22–27] for more details about users’ strategic behavior and game theory. As one of the most important functions of CRNs [4], spectrum access helps in preventing potential collisions among all users (including PUs and SUs) [28]. In this paper, we focus on the exclusive-use models (see [2, 6, 29]), namely, SUs transmit packets over a licensed spectrum only when no PU is using the spectrum. Therefore, PU can be considered to have higher priority over SUs. To characterize the endogenous priorities, CRNs can be modeled as a priority queueing system with heterogeneous users, in which the spectrum is considered as the server of this system. Numerous studies and surveys have been extensively carried out on CRNs, see, for example, [4, 9, 15, 30, 31], and the references therein. However, in previous literatures (see [11–14]), it is generally assume that PU has preemptive priority over SUs, i.e., an arriving PU request can push out the SU in service and occupy the spectrum. The interrupted SU will resume service once the spectrum is available again. In the context of communication protocol, it is natural to assume that each PU is given higher priority over SUs since PUs always are charged with service fees. Nevertheless, from a theoretic viewpoint, it is an interesting and challenging problem whether PU should preempt the SU in service in CR systems. In this paper, all PUs are considered to be symmetric, and we only study the case of a single PU who randomly 2

generates service requests. According to the IEEE 802.11 protocol, an arriving SU will enter a virtual waiting space (i.e., SU buffer) if he finds the PU band unavailable. Once the PU band is idle, the PU band immediately transmits SU packets. We consider whether PU should preempt the SU in service from the viewpoint of strategic behavior of users and focus on the throughput-maximization along with the social welfare maximization of the CR system. To this end, a CRN is modeled as a preemptive priority queueing system and a nonpreemptive priority queueing system, respectively. From the viewpoints of the throughput and the social welfare of the system, we make a comparison and determine which case is better. Beyond all doubts, users’ strategic decisions have vital impacts on the system performance. To explore whether preemptions should be permitted, we need to consider users’ equilibrium joining strategies in the nonpreemptive and preemptive priority queueing systems. After obtaining users’ equilibrium joining strategies in the two cases, we compare the throughput and the social welfare correspondingly and then indicate which case is better: to preempt or not to preempt. To the best of our knowledge, users’ equilibrium strategic behavior in the nonpreemptive case has not been discussed in the existing literature, while the preemptive case has been studied in [32] when the queue length of the system is observable to strategic users. Recently, [13] considered users’ equilibrium joining strategies in a preemptive priority queueing system with different information levels. Our setting contributes to the game-theoretic analysis of CR systems as well as queueing-game applications in several ways: (1) In the nonpreemptive case, we carry out the game-theoretic analysis on users’ strategic behavior and derive two dimensional equilibrium joining strategies including dominant strategy, pure strategy and mixed strategy. (2) From the viewpoint of throughput maximization, we find that there exists a threshold such that if ν1 defined in Section 2 is greater than the threshold, preemptions reduce the throughput of the system and should be prohibited. On contrary, if it is smaller, preemptions should be encouraged. (3) From the viewpoint of social welfare, we find that the social welfare corresponding to the preemptive case is always less than that corresponding to the nonpreemptive case. Therefore, to maximize the social welfare, for the social planner, PU should be prohibited to preempt the SU in service. The rest of the paper is organized as follows: Section 2 presents our model and gives some notations. In Section 3, we consider users’ equilibrium joining strategic behavior and derive dominant strategy, pure strategy and mixed strategy in the nonpreemptive case. For the preemptive case, we also summarize users’ equilibrium joining strategy in this section. Section 4 discusses whether PU should be encouraged to preempt the SU in service from two different perspectives. Finally, conclusions and future works are given in Section 5. 2. Model Description In this paper, we consider a cognitive radio system with a single band. The PU band is licensed to a PU who randomly sends service requests. We assume that the inter-arrival time of PU requests is exponential with rate λ1 . The PU band also can be flexibly accessed by SUs if it is not occupied by the PU. PU has priority over SUs. The inter-arrival time of SUs is assumed to follow the exponential distribution with rate λ2 . The PU requests and the SU 3

To preempt or not to preempt?

PU packet SU packet

PU band lower priority

Departure

higher priority

Figure 1: Illustration of the service mechanism of PUs and SUs. Yes New arriving Request

The PU request is lost

No

Does the PU decide to enter the system ?

Yes

The PU request lines up in the system

No

Is it the PU request ?

Yes

PU request

Is there any other PU request lining before the PU?

No Yes

Is the PU band idle or used by a SU?

The PU request is transmitted

No The SU request is transmitted

SU request No

Yes Is the PU band idle?

The PU band is used by the SU

Is there a Yes PU request needing the PU band?

PU is using the PU band and the SU is waiting

The PU request is transmitted

No No Does the SU decide to enter the system? No

Yes

The SU lines up in thesystem

The SU request is lost

Is there any other SU lining up before the SU? Yes

Figure 2: The network topology for the preemptive case.

requests are served according to the order of arrivals, respectively. The PU band can transmit one SU/PU packet at a time. If a request cannot be served upon arrival, it is lined up before the server. Based on the statement above, we model the CR system as a priority queue system. The illustration about the service mechanism of PU requests and SU requests is given in Figure 1. An important concern is about whether PU should preempt the SU in service. Therefore, in this paper, we consider two different service mechanisms, namely, preemptive case and nonpreemptive case. The former denotes that an arriving PU request can push out the SU in service while the latter means that no preemption is permitted. The related network topologies are showed in Figures 2-3. Some assumptions and notations are given here. The time each PU (or SU) request occupies the PU band is called the service time of PU (or SU), and assumed to be exponential with rate µ1 (or µ2 ). After each service of PU (or SU) request, the user will receive the reward of R1 (or R2 ). We assume that f1 (x) and f2 (x) denote the waiting costs 4

The PU request is transmitted

The PU request is lost

Yes

Yes

No

No Is the PU band idle?

PU request

No

Does the PU decide to enter the system ?

Yes

Is there any other PU request lining before the PU request?

The PU request lines up in the system

Yes

Is it the PU request ?

No

SU request

Is the PU band idle?

Yes

The SU request is transmitted No

No

New arriving Request

Yes Does the SU decide to enter the system ? No

The SU request lines up in the sytem

The SU request is lost

Is there any other SU lining before the SU?

No

Is there any PU request in the system?

Yes Yes

Figure 3: The network topology for the non-preemptive case.

of PU and SU per x time units, respectively. f1−1 (x) and f2−1 (x) are their inverse functions. As in Guo and Zipkin [33], we also assume that the waiting cost functions are increasing, continuous and positive. For simplicity, denote by ν1 = f1−1 (R1 ) and ν2 = f2−1 (R2 ) throughout this paper. In practice, neither PUs nor SUs can observe the actual queue lengths (including the queue length of PUs and the queue length of SUs). Therefore, we suppose that all information about the queue lengths is concealed. Also assume that parameters λ1 , λ2 , µ1 , µ2 , ν1 , ν2 , R1 , R2 are known to all users. Since the users are heterogeneous in this paper, we get a two-dimensional users’ strategy. A concept “strategy (q1 , q2 )” from game theory, which denotes that an arriving PU request is sent to the system with probability q1 and an arriving SU joins the system with probability q2 , will be frequently mentioned in this paper. For the convenience of narration, without causing confusion “PU request” is called PU in short in the rest of the paper. For example, hereinafter “ an arriving PU” denotes an arriving PU request. An arriving PU (or SU) will join the system if q1 = 1 (or q2 = 1), and balk if q1 = 0 (or q2 = 0). Strategy (q1 , q2 ) is called pure strategy if qi = 0 or 1, i = 1, 2; otherwise, it is a mixed strategy.

5

3. Equilibrium Strategic Behaviors In this section, we consider users’ equilibrium joining strategic behaviors in the preemptive and nonpreemptive cases, respectively. Specifically, for the nonpreemptive case, we obtain dominant joining strategy, pure joining strategy and mixed joining strategy, which are showed in Theorems 3.2-3.4, respectively; for the preemptive case, users’ equilibrium joining strategic behavior has been studied in [13], and we summarize the related results in Section 3.2. To avoid the situation that all PUs reject to join the system upon arrivals, we assume that ν1 > 1/µ. Under this condition, if an arriving PU faces an idle system, his expected net benefit will be positive, i.e., R1 − f1 (1/µ) > 0. That

means the equilibrium joining probability of PUs is nonzero no matter whether preemptions are permitted or not. 3.1. Nonpreemptive Case

First we consider the nonpreemptive case. Let W1 (q1 , q2 ) and W2 (q1 , q2 ) be the expected sojourn times of PUs and SUs if users adopt the strategy (q1 , q2 ), respectively. According to [34], the expected sojourn times of PUs and SUs can be written as

µ2 ρ1 q1 + µ1 ρ2 q2 1 + , µ1 µ2 (1 − ρ1 q1 ) µ1

(1)

µ2 ρ1 q1 + µ1 ρ2 q2 1 + . µ1 µ2 (1 − ρ1 q1 )(1 − ρ1 q1 − ρ2 q2 ) µ2

(2)

W1 (q1 , q2 ) = and W2 (q1 , q2 ) =

We easily find that the first order derivative of W1 (q1 , q2 ) with respect to q1 and the first order derivative of W2 (q1 , q2 ) with respect to q2 are positive, that is,

and

∂W1 (q1 , q2 ) ρ1 (µ2 + q2 µ1 ρ2 ) = > 0, ∂q1 µ1 µ2 (1 − q1 ρ1 )2

(3)

∂W2 (q1 , q2 ) q1 µ2 ρ1 ρ2 + µ1 (1 − q1 ρ1 )ρ2 > 0, = ∂q2 µ1 µ2 (1 − q1 ρ1 )(1 − q1 ρ1 − q2 ρ2 )2

(4)

which mean that the expected sojourn time of PUs (or SUs) is increasing in q1 (or q2 ). This result will be used in the rest of the paper and we summarize it in the following lemma. Lemma 3.1. W1 (q1 , q2 ) is increasing in q1 for a fixed q2 , and W2 (q1 , q2 ) increases with q2 for a fixed q1 . The above result is quite intuitive. As q1 (or q2 ) grows, more PUs (or SUs) join the system, so the system becomes more congested, and then PUs (or SUs) need to spend longer expected sojourn time. Upon arrival, a user (PU or SU) needs to decide whether to join the system or not. His decision depends on the available information. The information about the queue lengths of PUs and SUs is unobservable, so the user can make decision only based on available information including the values of λ1 , λ2 , µ1 , µ2 , ν1 , ν2 , R1 , R2 . An arriving user evaluates which strategy can produce more expected net benefit, and then chooses the best one. The expected net benefit equals the reward after receiving the service minus the waiting cost. If all other users join the system 6

with strategy (q1 , q2 ) and an arriving PU also joins the system, the arrival will receive the reward of R1 and spend the waiting cost of f1 (W1 (q1 , q2 )), then his expected net benefit is R1 − f1 (W1 (q1 , q2 )), which is denoted by S 1 (q1 , q2 ), i.e., S 1 (q1 , q2 ) , R1 − f1 (W1 (q1 , q2 )). By using the similar analysis, the expected net benefit of an arriving SU is

R2 − f2 (W2 (q1 , q2 )). Let S 2 (q1 , q2 ) , R2 − f2 (W2 (q1 , q2 )). If the expected net benefit corresponding to an arriving PU (or SU) is positive, joining the system is the best decision for him; if less than zero, balking is the best; if equals zero, he is indifferent between joining and balking. Let (qe1 , qe2 ) be users’ equilibrium joining strategy, which means that PUs join the system with qe1 and SUs join the system with qe2 in a Nash equilibrium situation. Now we first consider an extreme case, i.e., dominant strategy, in which an arriving user can obtain more net benefit than any other strategy. Theorem 3.2. If ν1 ≥ ν˜ 1 (µ1 , µ2 , ρ1 , ρ2 ) and ν2 ≥ ν˜ 2 (µ1 , µ2 , ρ1 , ρ2 ) where µ2 ρ1 + µ1 ρ2 1 + , µ1 µ2 (1 − ρ1 ) µ1

(5)

µ2 ρ1 + µ1 ρ2 1 + , µ1 µ2 (1 − ρ1 )(1 − ρ1 − ρ2 ) µ2

(6)

ν˜ 1 (µ1 , µ2 , ρ1 , ρ2 ) = and ν˜ 2 (µ1 , µ2 , ρ1 , ρ2 ) =

then strategy (1, 1), i.e., joining the system for all arriving users, is a dominant equilibrium. Proof: Equations (1)-(2) give the expected sojourn times of PUs and SUs, W1 (q1 , q2 ) and W2 (q1 , q2 ), respectively. Take (q1 , q2 ) = (1, 1), then W1 (1, 1) = ν˜ 1 (µ1 , µ2 , ρ1 , ρ2 ) and W2 (1, 1) = ν˜ 2 (µ1 , µ2 , ρ1 , ρ2 ). According to the condition of Theorem 3.2, we obtain W1 (1, 1) ≤ ν1 and W2 (1, 1) ≤ ν2 . We easily get R1 − f1 (W1 (1, 1)) ≥ 0 and R2 − f2 (W2 (1, 1)) ≥ 0

since ν1 = f1−1 (R1 ), ν2 = f2−1 (R2 ), and f (x) is an increasing function. For an arbitrary joining strategy (qˆ 1 , qˆ 2 ), according to Lemma 3.1, the following equations hold: R1 − f1 (W1 (qˆ 1 , qˆ 2 )) ≥ R1 − f1 (W1 (1, 1)) ≥ 0,

(7)

R2 − f2 (W2 (qˆ 1 , qˆ 2 )) ≥ R2 − f2 (W2 (1, 1)) ≥ 0.

(8)

The left side of (7) is the expected net benefit of an arriving PU if he joins the system and other users adopt strategy (qˆ 1 , qˆ 2 ) while the left side of (8) is the expected net benefit of an arriving SU. From (7) and (8), we have Ri − fi (Wi (qˆ 1 , qˆ 2 )) ≥ 0, i = 1, 2. Joining the system is the best response for any arriving user no matter what strategy other

users adopt. Hence, strategy (1, 1) is a dominant equilibrium.



Rational users must choose the dominant strategy because it is optimal to adopt such the strategy no matter what strategy other users choose. Note that the dominant strategy in Theorem 3.2 is also a pure strategy. Theorem 3.2 shows that a dominant pure strategy exists if ν1 , ν2 exceed the bounds, ν˜ 1 (µ1 , µ2 , ρ1 , ρ2 ) and ν˜ 2 (µ1 , µ2 , ρ1 , ρ2 ), respectively. To explain ν1 and ν2 more straightforward, we consider specific waiting cost functions f1 (x) = C1 x and f2 (x) = C2 x, which were used in the large number of literatures [32, 35–39]. Obviously, C1 , C2 are the waiting costs of PU and SU per time unit, respectively. In this case, νi = Ri /Ci , i = 1, 2. We easily find ν1 (or ν2 ) is the ratio of the reward of PU (or SU) to the waiting cost of PU (or SU), namely, ν1 (or ν2 ) is the income generated by per unit cost for a PU (or an 7

SU). Now we consider another case in which a pure strategy different from Theorem 3.2 exists. Theorem 3.3. If ν1 ≥

ρ1 µ1 (1−ρ1 )

+

1 µ1

and ν2 ≤

ρ1 µ1 (1−ρ1 )2

+

1 µ2 ,

(1, 0) is an equilibrium and it is a pure strategy.

Proof: If the condition of Theorem 3.3 holds, from (1) and (2) we have S 1 (1, 0) = R1 − f1 (W1 (1, 0)) ≥ 0,

(9)

S 2 (1, 0) = R2 − f2 (W2 (1, 0)) ≤ 0.

(10)

From (9), the expected net benefit of an arriving PU is nonnegative if he decides to join the system and all other users adopt strategy (1, 0), so joining the system is the best decision for the PU. From (10), we also find S 2 (1, 0) ≤ 0, so balking is an optimal decision for an arriving SU. Therefore, if all other users adopt strategy (1, 0), the best response

of an arriving user (PU or SU) is also to adopt the strategy. That is to say, strategy (1, 0) is an equilibrium. Obviously, 

it is also a pure strategy.

Theorem 3.3 shows that under certain conditions “join the system if an arriving user is a PU and balk if the arrival is an SU” is a pure equilibrium strategy. We will find that in other situations some mixed strategies exist. The corresponding results are summarized in the following theorem. ρ1 + µ11 and ν2 ≤ µ11 + ν1 (ν1 µ1 − 1), (qea , 0) is an equilibrium, where qea = Theorem 3.4. (a) If µ11 < ν1 < µ1 (1−ρ 1) 2 −µ1 ρ2 e (b) (qb , 1) is an equilibrium where qeb = µ1 µ2µν11µ−µ , if the following equations hold: 2 ν1 ρ1

1 1 ρ2 µ2 ρ1 + µ1 ρ2 + , + < ν1 < µ1 µ2 µ1 µ2 (1 − ρ1 ) µ1 and

µ1 ν1 −1 µ1 ν 1 ρ 1 .

(11)

µ2 ν1 (µ1 ν1 − 1) 1 + . µ2 µ1 ρ2 + µ2 (1 − µ1 ρ2 ν1 )

(12)

µ1 ν 1 − µ1 ν 2 + µ1 µ2 ν 1 ν 2 , µ1 µ2 ν1 ν2 ρ1 + µ2 ν2 ρ1 − µ1 ν2 ρ1

(13)

µ2 (ν1 + µ1 ν12 − ν2 ) . (µ1 − µ2 − µ1 µ2 ν1 )ν2 ρ2

(14)

ν2 ≥

(c) If 0 < qec , qed < 1, (qec , qed ) is an equilibrium, where qec = and qed = (µ1 −µ1 ρ1 +µ2 ρ1 )(µ2 ν2 −1) µ1 µ2 (µ2 ν2 +ρ1 −µ2 ν2 ρ1 (µ1 µ2 ν2 −µ1 )(1−ρ1 )2 −µ2 ρ1 e (µ1 µ2 ν2 −µ1 )(1−ρ1 )ρ2 +µ1 ρ2 and 0 < q f <

(d) If ν1 ≥

and

ρ1 µ1 (1−ρ1 )2

+

1 µ2

< ν2 < ν˜ 2 (µ1 , µ2 , ρ1 , ρ2 ), (1, qef ) is an equilibrium, where qef =

1.

Proof: (a) According to the condition of Theorem 3.4(a),

1 µ1

< ν1 <

ρ1 µ1 (1−ρ1 )

S 1 (1, 0) = R1 − f1 (W1 (1, 0)) = R1 − f1 8

1 µ1 ,

then

! 1 > 0, µ1

(15)

! ρ1 1 + < 0. µ1 (1 − ρ1 ) µ1

(16)

S 1 (0, 0) = R1 − f1 (W1 (0, 0)) = R1 − f1 and

+

By Lemma 3.1, W1 (q, 0) is increasing in q. From (15)-(16), S 1 (q, 0) = R1 − f1 (W1 (q, 0)) = 0, 0 < q < 1 has a unique solution. Through simple derivations, we can obtain the solution of the equation is qa . Therefore, we get S 1 (qa , 0) = R1 − f1 (W1 (qea , 0)) = 0. On the other hand, From (2), we have W2 (qea , 0) = we have

1 µ1

+ ν1 (ν1 µ1 − 1). By the known condition ν2 ≤

S 2 (qa , 0) = R2 − f2 (W2 (qea , 0)) ≤ 0.

(17) 1 µ1

+ ν1 (ν1 µ1 − 1), (18)

From (17) and (18), the expected net benefit of an arriving PU (or SU) is zero (or non-positive) if all other users adopt the strategy (qa , 0) and he decides to join the system, then the strategy (qa , 0) is also the best decision for the arrival. So (qa , 0) is an equilibrium. (b) The method similar to the proof of Theorem 3.4(a) is used here. From (11), we get R1 − f1 (W1 (0, 1)) > 0,

(19)

R1 − f1 (W1 (1, 1)) < 0.

(20)

According to (19), (20) and Lemma 3.1, R1 − f1 (W1 (q, 1)) = 0, 0 < q < 1 has a unique solution qb , i.e., R1 − f1 (W1 (qb , 1)) = 0. Hence, if other users choose the strategy (qeb , 1), it is also optimal to join the system with probability

qb for an arriving PU. From (12), S 2 (qb , 1) = R2 − f2 (W2 (qeb , 1)) ≥ 0. That means, if other users choose the strategy

(qeb , 1), the best response of an arriving SU is to join the system. In brief, the strategy (qeb , 1) is also optimal for an arriving user (PU or SU) if all other users choose the same strategy (qeb , 1). So the strategy (qeb , 1) is an equilibrium. (c) The strategy (qec , qed ) is an equilibrium where 0 < qec , qed < 1, if and only if     e e   R1 − f1 (W1 (qc , qd )) = 0,      R2 − f2 (W2 (qec , qed )) = 0.

(21)

The reason is given as follows. If other users choose the strategy (qec , qed ), the expected net benefit of an arriving user (no matter PU or SU) is zero. Then the user is indifferent between joining and balking. So the strategy (qec , qed ) is also optimal for the arriving user, that is to say, the strategy (qec , qed ) is an equilibrium. From (1) and (2), (21) can be rewritten as

    e e   W1 (qc , qd ) =      W2 (qec , qed ) =

µ2 ρ1 qec +µ1 ρ2 qed µ1 µ2 (1−ρ1 qec )

+

1 µ1

= ν1 ,

µ2 ρ1 qec +µ1 ρ2 qed µ1 µ2 (1−ρ1 qec )(1−ρ1 qec −ρ2 qed )

9

+

1 µ2

(22) = ν2 .

Then we have

    e e e   µ2 ρ1 qc + µ1 ρ2 qd = µ2 (ν1 µ1 − 1)(1 − ρ1 qc ),      µ2 ρ1 qec + µ1 ρ2 qed = µ1 (µ2 ν2 − 1)(1 − ρ1 qec )(1 − ρ1 qec − ρ2 qed ).

(23)

Solve the equations above, we get (13) and (14). Therefore, if 0 < qec , qed < 1, the strategy (qec , qed ) is an equilibrium. (d) We ignore the proof of Theorem 3.4(d) since it is similar to the proof of Theorem 3.4(b).



Remark 3.5. If ν1 is sufficiently large, the equilibrium joining probability of PUs equals one, i.e., qe1 = 1. The result can be found by observing Theorem 3.2, Theorem 3.3 and Theorem 3.4(d). Remark 3.6. According to Theorem 3.2 and Theorem 3.4(b), the equilibrium joining probability of SUs in the nonpreemptive case equals one (i.e., qe2 = 1) if one of the following two conditions holds: (1) ν1 ≥ ν˜ 1 (µ1 , µ2 , ρ1 , ρ2 ) and ν2 ≥ ν˜ 2 (µ1 , µ2 , ρ1 , ρ2 ); +µ1 ρ2 ν1 (µ1 ν1 −1) (2) µ11 + µρ22 < ν1 < µµ12µρ21(1−ρ + µ11 and ν2 ≥ µ12 + µ1 ρµ22+µ . 1) 2 (1−µ1 ρ2 ν1 ) Remark 3.7. The equilibrium joining probability of SUs in the nonpreemptive case equals zero (i.e., qe2 = 0) if one of the following two conditions holds: ρ1 + µ11 and ν2 ≤ µ11 + ν1 (ν1 µ1 − 1); (1) µ11 < ν1 < µ1 (1−ρ 1) ρ1 ρ1 1 + µ11 and ν2 ≤ µ1 (1−ρ (2) ν1 ≥ µ1 (1−ρ 2 + µ . 1) 2 1) This result can be easily found from Theorem 3.3 and Theorem 3.4(a). For the nonpreemptive case, no arriving SU is willing to join the system if one of these two conditions holds. 3.2. Preemptive Case In Section 3.1, PUs have nonpreemptive priorities over SUs. We obtain users’ strategic behavior including the dominant strategy, the pure equilibrium strategy and the mixed equilibrium strategy. In this section, we consider the preemptive case. Let (qepu , qesu ) be users’ equilibrium joining strategy with the same parameters λ1 , λ2 , µ1 , µ2 , ν1 , ν2 , pre pre R1 , R2 as that defined. For the preemptive case, W su (q pu , q su ) and W su (q pu , q su ) denote the expected sojourn times of

PUs and SUs respectively if all users adopt the strategy (q pu , q su ), and can be computed from the following equations (see White and Christie [40]):

1 , µ1 − λ1 q pu

(24)

ρ2 − ρ1 q pu ρ2 q su + ρ1 q pu ρ2 q su (µ2 /µ1 ) . λ2 q su (1 − ρ1 q pu )(1 − ρ1 q pu − ρ2 q su )

(25)

W pu (q pu , q su ) = and W su (q pu , q su ) =

According to [13], if the information about the queue lengths is concealed, there exists an equilibrium strategy (qepu , qesu ) such that “PUs join the system with probability qepu and SUs join the system with probability qesu ” is an equilibrium, where qepu

   µ1     λ1 − =     1,

1 λ1 ν1 ,

if

1 µ1

< ν1 <

if ν1 ≥

10

1 µ1 −λ1 ,

1 µ1 −λ1 ,

(26)

and

    0, if ν2 ≤ νˆ 1 (µ1 , λ2 , µ2 , ρ1 , ρ2 , qepu ),        qesu =  q, ¯ if νˆ 1 (µ1 , λ2 , µ2 , ρ1 , ρ2 , qepu ) < ν2 < νˆ 2 (µ1 , λ2 , µ2 , ρ1 , ρ2 , qepu ),         1, if ν2 ≥ νˆ 2 (µ1 , λ2 , µ2 , ρ1 , ρ2 , qepu ),

(27)

pre e in which q¯ is the solution to R2 − f2 (W su (q pu , q)) = 0 and

νˆ 1 (µ1 , λ2 , µ2 , ρ1 , ρ2 , qepu ) =

ρ2 − ρ1 qepu ρ2 + ρ1 qepu ρ2 (µ2 /µ1 )

,

(28)

νˆ 2 (µ1 , λ2 , µ2 , ρ1 , ρ2 , qepu ) =

ρ2 − ρ1 qepu ρ2 + ρ1 qepu ρ2 (µ2 /µ1 )

.

(29)

λ2 (1 − ρ1 qepu )2

λ2 (1 − ρ1 qepu )(1 − ρ1 qepu − ρ2 )

Remark 3.8. According to (26) and (27), we can obtain the following results: 1 (a) If ν1 ≥ µ1 −λ and ν2 ≤ νˆ 1 (µ1 , λ2 , µ2 , ρ1 , ρ2 , qepu ), the strategy (1, 0) (i.e., joining the system for an arriving PU 1 and balking for an arriving SU) is a pure equilibrium strategy; 1 (b) If ν1 ≥ µ1 −λ and ν2 ≥ νˆ 2 (µ1 , λ2 , µ2 , ρ1 , ρ2 , qepu ), the strategy (1, 1) is an equilibrium strategy. It is the best 1 decision for an arriving user to join the system, then it is a pure strategy; (c) Otherwise, there exists a mixed strategy (qepu , qesu ) which is determined by (26) and (27). The results in Remark 3.8 are showed in Table 1. (1, 0) and (1, 1) are pure strategies while (1, qesu ), (qepu , 0), (qepu , 1) and (qepu , qesu ) are mixed strategies. Table 1 Pure strategies and mixed strategies

1 ν1 ≥ µ −λ 1 1 otherwise

ν2 ≤ νˆ 1 (µ1 , λ2 , µ2 , ρ1 , ρ2 , qepu ) (1, 0) (qepu , 0)

ν2 ≥ νˆ 2 (µ1 , λ2 , µ2 , ρ1 , ρ2 , qepu ) (1, 1) (qepu , 1)

otherwise (1, qesu ) (qepu , qesu )

Remark 3.9. Similar to Remark 3.5, for the preemptive case, the equilibrium joining probability of PUs equals one (i.e., qepu = 1) if ν1 is sufficiently large. The result can be obtained from (26). Remark 3.10. From (27), the equilibrium joining probability of SUs in the preemptive case equals one (i.e., qesu = 1) if ν2 is sufficiently large (specifically, ν2 is not less than νˆ 2 (µ1 , λ2 , µ2 , ρ1 , ρ2 , qepu )). Remark 3.11. The equilibrium joining probability of SUs in the preemptive case equals zero (i.e., qesu = 0) if ν2 is sufficiently small (specifically, ν2 is less than and equals νˆ 1 (µ1 , λ2 , µ2 , ρ1 , ρ2 , qepu )). This result can be easily found by (27). 4. Comparison between Preemptive and Nonpreemptive Cases In CRNs, throughput and social welfare are two important concerns. The size of the former embodies the efficiency of the system while the social welfare reflects the total net benefit of all users. From the viewpoint of the managers of CRNs, they hope to maximize the efficiency of the system. However, from the viewpoint of social planners, their objective is to maximize the overall social welfare. In this section, we consider whether or not PUs should preempt 11

the SU in service. From the perspective of throughput maximization, we find in some situations preemptions are more beneficial. But from the perspective of social welfare maximization, the nonpreemptive case always is better than the preemptive case. 4.1. Throughput Let λnon and λepre be the throughputs in the nonpreemptive and preemptive cases, respectively. According to the e queue theory, we know that throughput is identical to the total effective arrival rate of users. In the nonpreemptive case, (qe1 , qe2 ) is users’ equilibrium strategy, which has been given in Section 3.1. Then the throughput λnon is the e effective arrival rate of PUs plus the effective arrival rate of SUs, namely, λnon = λ1 qe1 + λ2 qe2 . e

(1)

For the preemptive case, (qepu , qesu ) is users’ equilibrium strategy, where qepu , qesu are determined by (26) and (27). Then the throughput λepre is given as follows: λepre = λ1 qepu + λ2 qesu .

(2)

If PUs have preemptive priorities over SUs, an arriving PU can preempt the SU in service. However, if PUs are given nonpreemptive priorities over SUs, an arriving PU can not push out the SU in service. Obviously, the expected sojourn time of a joining PU in the preemptive case is less than that in the nonpreemptive case. Therefore, PUs in the preemptive case are more willing to join the system than ones in the nonpreemptive case. On the other hand, since preemptions increase the service time of SU in service, SUs in the preemptive case have less willingness to join the system than ones in the nonpreemptive case. As we know, equilibrium joining probability is positively related to users’ joining willingness. Therefore, qepu > qe1 if 0 < qepu < 1; qe2 > qesu if 0 < qe2 < 1. As stated in the first paragraph in Section 3, the equilibrium joining probability of PUs in the preemptive case qepu must be nonzero. According to Remark 3.5 and Remark 3.9, qepu = qe1 = 1 if ν1 is sufficiently large. If qe2 = 0, all SUs in the nonpreemptive case are not willing to join the system. Then all SUs in the preemptive case also balk since preemptions make SUs spend longer expected waiting time in the system. That means qesu = qe2 = 0 if qe2 = 0. Based on the similar analysis, we have qesu = qe2 = 1 if qesu = 1. We summarize the results in the following remark. Remark 4.1. The equilibrium joining probability of PUs in the preemptive case always is more than and equals that in the nonpreemptive case, and the equilibrium joining probability of SUs in the preemptive case always does not exceed that in the nonpreemptive case. That is to say, qe1 ≤ qepu , and qe2 ≥ qesu . If 0 < qe2 < 1, qe2 > qesu . If 0 < qepu < 1, qepu > qe1 . Figure 4 shows the relation between equilibrium joining probabilities and ν1 for λ1 = 0.45, λ2 = 0.2, µ2 = 0.8, where Figure 4(a) is the case of ν2 = 5 and Figure 4(b) is the case of ν2 = 8. Figure 4 shows when ν1 is sufficiently large, the equilibrium joining probabilities of PUs equal one no matter whether PUs preempt the SU in service or not. This phenomenon has been discussed in Remark 3.5 and Remark 3.9. In addition, Figure 4(a) and Figure 4(b) 12

1 0.8

1 e qpu

0.8

e qpu

q2e 0.6

0.6

q1e

e qsu

q2e

0.4

q1e

0.4 e qsu

0.2

0.2 0

2.5

ν1 (ν2 = 5)

3

3.5

2.5

3

3.5

ν1 (ν2 = 8)

(a)

(b)

Figure 4: Relation between equilibrium joining probabilities and ν1 for λ1 = 0.45, λ2 = 0.2, µ2 = 0.8, µ2 = 1.

represent two different cases. The first is the situation that the equilibrium joining probabilities of SUs tend to zero as ν1 grows, but the second is the situation that the equilibrium joining probabilities of SUs tend to a positive value. We also find qe1 ≤ qepu and qe2 ≥ qesu under these two situations. Compared with the preemptive case, PUs (or SUs) join

the system with lower (or higher) probability in the nonpreemptive case. This is because preemptions are beneficial to PUs but adverse to SUs, so PUs (or SUs) have more (or less) willingness to join the system if preemptions are permitted. To answer whether or not permitting preemptions is a better policy from the perspective of throughput, we need to compare the throughput in the preemptive case with that in the nonpreemptive case. Theorem 4.2. (a) Consider the case of qepu = qe1 . The nonpreemptive case is not worse than the preemptive case from the viewpoint of throughput maximization. Specially, if the condition of Remark 3.7 holds, there is no difference between the preemptive case and the nonpreemptive case from the perspective of throughput maximization. (b) Consider the case of qepu , qe1 . From the viewpoint of throughput maximization, the nonpreemptive case is better than the preemptive case if λλ21 < difference between these two cases.

qe2 −qesu qepu −qe1 ;

the preemptive case is better if

λ1 λ2

>

qe2 −qesu qepu −qe1 ;

otherwise, there is no

= λ2 (qesu − qe2 ). According to Remark 4.1, λepre ≤ λnon Proof: (a) If qepu = qe1 , from (1) and (2) we have λepre − λnon e . e

Then the nonpreemptive case is not worse than the preemptive case from the viewpoint of throughput maximization. In addition, if the condition of Remark 3.7 holds, qe2 = 0. By Remark 4.1, qe2 = qesu = 0, then λepre = λnon e , that is, there is no difference between the preemptive case and the nonpreemptive case.

(b) We can easily obtain qepu > qe1 according to Remark 4.1 and the condition of qepu , qe1 . Comparing the corresponding to the nonpreemptive throughput λepre corresponding to the preemptive case with the throughput λnon e 

case, we immediately get Theorem 4.2.

Theorem 4.3. From the viewpoint of throughput maximization, the nonpreemptive case is not worse than the preemp13

tive case if

(

1 ν1 ≥ max ν˜ 1 (µ1 , µ2 , ρ1 , ρ2 ), µ1 − λ1

)

, B(µ1 , µ2 , ρ1 , ρ2 , ν1 , ν2 ).

Specially, the nonpreemptive case is better than the preemptive case if ν1 ≥ max

(

ρ1 µ1 (1−ρ1 )2

+

1 µ2

(3)

< ν2 < ν˜ 2 (µ1 , µ2 , ρ1 , ρ2 ) and

) 1 (µ1 − µ1 ρ1 + µ2 ρ1 )(µ2 ν2 − 1) , . µ1 µ2 (µ2 ν2 + ρ1 − µ2 ν2 ρ1 µ1 − λ 1

(4)

Proof: From (1) and (5), we get ρ1 1 + , µ1 (1 − ρ1 ) µ1 (µ1 − µ1 ρ1 + µ2 ρ1 )(µ2 ν2 − 1) , 0 < qef < 1, W1 (1, qef ) = µ1 µ2 (µ2 ν2 + ρ1 − µ2 ν2 ρ1 W1 (1, 0) =

W1 (1, 1) = ν˜ 1 (µ1 , µ2 , ρ1 , ρ2 ).

(5) (6) (7)

According to Lemma 3.1, we have W1 (1, 0) < W1 (1, qef ) < W1 (1, 1), that is ρ1 1 (µ1 − µ1 ρ1 + µ2 ρ1 )(µ2 ν2 − 1) + < < ν˜ 1 (µ1 , µ2 , ρ1 , ρ2 ). µ1 (1 − ρ1 ) µ1 µ1 µ2 (µ2 ν2 + ρ1 − µ2 ν2 ρ1

(8)

If ν1 ≥ B(µ1 , µ2 , ρ1 , ρ2 , ν1 , ν2 ), we get 1 (µ1 − µ1 ρ1 + µ2 ρ1 )(µ2 ν2 − 1) ρ1 + < < ν˜ 1 (µ1 , µ2 , ρ1 , ρ2 ) ≤ ν1 . µ1 (1 − ρ1 ) µ1 µ1 µ2 (µ2 ν2 + ρ1 − µ2 ν2 ρ1 If ν1 ≥ B(µ1 , µ2 , ρ1 , ρ2 , ν1 , ν2 ) and ν2 <

and

ρ1 µ1 (1−ρ1 )2

+

1 µ2

ρ1 + 1 , from Theorem 3.3, (qe1 , qe2 ) µ1 (1−ρ1 )2 µ2

(9)

= (1, 0); if ν1 ≥ B(µ1 , µ2 , ρ1 , ρ2 , ν1 , ν2 )

< ν2 < ν˜ 2 (µ1 , µ2 , ρ1 , ρ2 ), from Theorem 3.4(d), (qe1 , qe2 ) = (1, qef ); if ν1 ≥ B(µ1 , µ2 , ρ1 , ρ2 , ν1 , ν2 )

and ν2 ≥ ν˜ 2 (µ1 , µ2 , ρ1 , ρ2 ), from Theorem 3.2, (qe1 , qe2 ) = (1, 1). Therefore, for an arbitrary ν2 , qe1 = 1 if ν1 ≥

B(µ1 , µ2 , ρ1 , ρ2 , ν1 , ν2 ). By Remark 4.1, we immediately obtain qe1 = qepu = 1 and qe2 ≥ qesu since qe1 = 1. Then λepre = λ1 qepu + λ2 qesu ≤ λ1 qe1 + λ2 qe2 = λnon e . That means the nonpreemptive case is not worse than the preemptive case if ν1 ≥ B(µ1 , µ2 , ρ1 , ρ2 , ν1 , ν2 ).

Now we consider the special case. If

ρ1 µ1 (1−ρ1 )2

equations hold: ν1 ≥

+

1 µ2

< ν2 < ν˜ 2 (µ1 , µ2 , ρ1 , ρ2 ) and (4) is satisfied, the following

(µ1 − µ1 ρ1 + µ2 ρ1 )(µ2 ν2 − 1) , µ1 µ2 (µ2 ν2 + ρ1 − µ2 ν2 ρ1

(10)

1 . µ1 − λ1

(11)

and ν1 ≥

According to Theorem 3.4(d) and (10), we have qe1 = 1 and 0 < qe2 < 1. From (11) and (26), we get qepu = 1. Since 0 < qe2 < 1, By Remark 4.1, qe2 > qesu . Therefore, λepre = λ1 + λ2 qesu < λ1 + λ2 qe2 = λnon e , namely, the nonpreemptive case is better than the preemptive case. This completes the proof. 14



0.7 0.6

T hroughput

T hroughput

0.5

0.4

0.3

2.5

3

0.4 0.3

P reemptive case N onpreemptive case 0.2

0.5

0.2

3.5

ν1 (ν2 = 5)

P reemptive case N onpreemptive case 2.5

3

3.5

ν1 (ν2 = 8)

(a)

(b)

Figure 5: Relation between total throughput and ν1 for λ1 = 0.45, λ2 = 0.2, µ2 = 0.8, µ2 = 1.

Theorem 4.3 shows if ν1 is greater than the threshold B(µ1 , µ2 , ρ1 , ρ2 , ν1 , ν2 ), prohibiting any preemption is a better decision from the perspective of throughput maximization. In addition to the theoretical analysis above, some numerical examples are given as follows. Figure 5(a) implies that the preemptive case is better than the nonpreemptive case as ν1 is small while the nonpreemptive case is not worse than the preemptive case as ν1 is sufficiently large. Figure 5(b) shows that there exists a constant number such that the preemptive case is better than the nonpreemptive case if ν1 is less than the value. This phenomenon is the same to Figure 5(a), but different from Figure 5(a), the nonpreemptive case is strictly better than the preemptive case if more than the value. Recalling Figure 4, as ν1 grows, the equilibrium joining probability of PUs tends to one no matter whether PUs preempt the SU in service or not, but the equilibrium joining probabilities of SUs in the preemptive and nonpreemptive cases may tend to zero or nonzero. For λ1 = 0.45, λ2 = 0.2, µ2 = 0.8, µ2 = 1, ν2 = 5, if ν1 is sufficiently large, the equilibrium joining strategy is (1, 0) no matter whether preemptions are permitted or not (see Figure 4(a)), so in this situation the throughput corresponding to the preemptive case is identical to that corresponding to the nonpreemptive case (see Figure 5(a)). For λ1 = 0.45, λ2 = 0.2, µ2 = 0.8, µ2 = 1, ν2 = 8, the equilibrium joining probabilities of SUs in the preemptive and nonpreemptive cases are nonzero and qe2 > qesu as ν1 is sufficiently large (see Figure 4(b)). Then λepre = λ1 + λ2 qesu < λ1 + λ2 qe2 = λnon e . Therefore, the throughput corresponding to the nonpreemptive case is strictly greater than that corresponding to the preemptive case (see Figure 5(b)). 4.2. Social Welfare In this section, we consider whether PUs should be given preemptive priorities from the perspective of the social planner. The social planner always maximizes the total expected benefit of all users (called social welfare for short) by adopting an optimal policy. If the preemptive case can produce more total expected benefit than the nonpreemptive case, preemptions should be permitted; otherwise, the nonpreemptive case is better. 15

Let S W non and S W pre be the social welfares per time unit under the nonpreemptive and preemptive cases, respectively. For the nonpreemptive case, the social welfare per time unit comes from two parts. The first is the total expected net benefit of all joining PUs per time unit and the second is the total expected net benefit of all joining SUs per time unit. The former equals the effective arrival rate λ1 qe1 times the expected net benefit of each joining PU R1 − f1 (W1 (qe1 , qe2 )). Similarly, the latter is λ2 qe2 [R2 − f2 (W2 (qe1 , qe2 ))]. Therefore, the social welfare per time unit is

computed from:

S W non = λ1 qe1 [R1 − f1 (W1 (qe1 , qe2 ))] + λ2 qe2 [R2 − f2 (W2 (qe1 , qe2 ))].

(12)

By the similar analysis, in the preemptive case, the social welfare per time unit can be written as S W pre = λ1 qepu [R1 − f1 (W1pre (qepu , qesu ))] + λ2 qesu [R2 − f2 (W2pre (qepu , qesu ))].

(13)

If S W pre > S W non , it is an optimal decision to permit preemptions. On contrary, if S W pre < S W non , the social planner should perform the nonpreemptive policy. The social welfare corresponding to the preemptive case is compared with that corresponding to the nonpreemptive case in Figure 6. Intuitively, if PU after being served receives a very high reward, preemptions might be beneficial to the social welfare. However, numerical examples show that the intuitive understanding is wrong. Figures 6(a)-(d) are the cases of R1 = 6, 10, 24, 200, respectively. Red solid lines denote the social welfare in the preemptive case while blue dash lines are the social welfare in the nonpreemptive case. No matter R1 equals 6, 10, 24 or 200, the blue dash line is always above the red solid line. So Figure 6 shows that the social welfare corresponding to the nonpreemptive case is always greater than that corresponding to the preemptive case. Therefore, from the viewpoint of the social planner, it is an optimal decision to prohibit any preemption. 5. Conclusions In this paper we considered a CRN with a single primary user and multiple secondary users. Two types of priority queueing systems were used to model the CR system. We first studied users’ strategic behavior and obtained two dimensional equilibrium joining strategies. Our main objective was to explore whether or not preemptions should be permitted. We discussed this problem from the perspectives of the throughput and the social welfare of the system. Comparing the throughput corresponding to the preemptive case with that corresponding to the nonpreemptive case, we found the preemptive case is better if ν1 is small. If ν1 exceeds the threshold B(µ1 , µ2 , ρ1 , ρ2 , ν1 , ν2 ), prohibiting any preemption is an optimal decision from the perspective of the throughput. For the social planner, his aim is always to maximize the total expected net benefit. Numerical examples showed that the social welfare in the nonpreemptive case is always higher than that in the preemptive case. Therefore, it is an optimal decision to prohibit any preemption from the viewpoint of the social welfare. For the future work, one could extend the current model to the case of multiple types of users and multiple PU bands. In such a scenario, users’ strategic behavior and optimal control of the system are two interesting concerns. Multi-player game among different types of users deserves future research. In 16

5

Social welf ares

Social welf ares

3

2

1

1

3 2 1

P reemptive case N onpreemptive case

0

4

0

2

P reemptive case N onpreemptive case 1

µ1 (R1 = 6)

1.5

2

µ1 (R1 = 10)

(a)

(b)

12 90

Social welf ares

Social welf ares

10 8 6 4

82

P reemptive case N onpreemptive case

2 0

86

1

1.5

P reemptive case N onpreemptive case 78

2

µ1 (R1 = 24)

1

1.5

2

µ1 (R1 = 200)

(c)

(d)

Figure 6: Comparison between the social welfare corresponding to the preemptive case and the social welfare corresponding to the nonpreemptive case for λ1 = 0.45, λ2 = 0.2, µ2 = 1, C1 = 4, R1 = 6, C2 = 1, R2 = 5. (a) R1 = 6, (b) R1 = 10, (c) R1 = 24, (d) R1 = 200.

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