A Game-Theoretic Analysis of Bandwidth Allocation under a User-Grouping Constraint

Abstract

A new bandwidth allocation model is studied in this paper. In this model, a system, such as a communication network, is composed of a finite number of users, and they compete for limited bandwidth resources. Each user adopts the decision that maximizes his or her own benefit characterized by the utility function. The decision space of each user is subject to constraints. In addition, some users form a group, and their joint decision space is also subject to constraints. Under the assumption that each user’s utility function satisfies some continuity and concavity conditions, the existence, uniqueness, and fairness, in some appropriate sense, of the Nash equilibrium point in the allocation game are proved. An algorithm yielding a sequence converging to the equilibrium point is proposed. Finally, a numerical example with detailed analysis is provided to illustrate the effectiveness of our work.

1. Introduction

With the widespread use of internet and the increasing popularity of mobile devices, more and more people can get online at almost anytime and anywhere. An immediate challenge facing the significant increment of online users is the support of quality of service (QoS). Over the past decade considerable efforts have been made to ensure the smooth operation of the networking systems. For example, the load balancing problems were considered by Anselmi et al. [1] and Ayesta et al. [2]. The routing problems were studied by La and Anantharam [3], Richman and Shimkin [4], Boulogne et al. [5], and Korilis et al. [6–8]. Niyato and Hossain [9] studied the practical issue such as the admission control for the wireless broadband standard. Ganesh et al. [10] and Yaïche et al. [11] considered the pricing issues. In modeling the networking problems, quite often the bandwidth availability is the main concern and has its role in the associated performance measure. The network system quantifies the results caused by different operation scenarios and seeks the approach leading to the greatest benefits in some sense. Since the benefit of any network user inevitably involves that of other users, its evaluation is mostly carried out in the context of game theory. In the survey paper Altman et al. collected a long list of networking models based on game theoretic formulation. Interested readers are referred to [12] and the rich reference therein.

As mentioned above, the bandwidth availability is the major concern in many networking problems. The bandwidth allocation is thus the core issue as far as the quality of service is concerned. While the bandwidth allocation problem was considered by many authors in different contexts of networking protocols or communication standards (see e.g., [9, 11, 13, 14]), at a high level of abstraction the problem can be regarded as the classical resources distribution problem studied in many professional fields such as economics, management science, and operations research. Given a finite number of units competing for the limited resources, how does each unit decide its share based on its own utility function? Lazar et al. [15] formulated this problem for the network composed of noncooperative users. Under certain monotonicity, differentiability, and convexity assumptions on the cost function the unique existence and certain fairness property of the Nash equilibrium point (NEP) were proved. An algorithm based on Gauss-Seidel and Jacobi schemes was proposed and proved to yield a sequence converging to the NEP. However, the framework in [15] assumes only the natural constraint for the bandwidth allocation. That is, the feasible bandwidth of any user falls within the interval lower bounded by zero and upper bounded by the bandwidth available for that user. In practice some techniques such as the bandwidth throttling and bandwidth/traffic shaping [16] are available to provide more adaptive bandwidth control. To address this issue, Rhee and Konstantopoulos [17, 18] relaxed the assumption and allowed some prespecified numbers for the upper and lower bounds of the bandwidth. This relaxation increases the flexibility of flow control and helps the QoS satisfaction by the networking system. On the other hand, in modern broadband communication systems some users might form a group and expect the group-wise QoS, in addition to the user-wise one. For example, the customers of an internet service provider (ISP) might include the individuals and a company with many employees. To maintain the QoS, parameters would be assigned to bound the bandwidth of each individual and each employee. Furthermore the ISP and the company would set the constraint for the employee-averaged, or equivalently, employee-totaled bandwidth, as shown in Figure 1. A similar concept of group constraint can be seen in the cost-effective broadband access network such as the Ethernet-based passive optical network (EPON) [19, 20]. This system is composed of an optical line terminal (OLT) and many optical network units (ONUs). The OLT is situated in the central office and ONUs are distributed over the remote areas for multimedia communication with the subscribers. In the upload process each ONU adopts the time division multiplexing access (TDMA) protocol to transmit data frames to the OLT, in the sense that each ONU only transmits the data during the time slots specifically scheduled for it [21]. The protocol avoids the frame collisions between different ONUs at the cost of imposing the upper bound for the time-average flow of each ONU and the upper bound for the total flow of all ONUs.

Figure 1 

The system bandwidth allocation 𝐫=(𝑟1,𝑟2,…,𝑟𝑛) subject to user-wise constraints: 𝑟𝑖≤𝑟𝑖≤𝑟𝑖 for 𝑖∈{1,2,…,𝑛}, and a user-grouping constraint 𝑅𝑔≤∑𝑚𝑖=1𝑟𝑖≤𝑅𝑔. The total allocation 𝑅=∑𝑛𝑖=1𝑟𝑖 is less than 𝑇, the total bandwidth.

In light of the bandwidth sharing mechanism in EPON and other similar systems, we extend the existing results to include the user-grouping constraint for better model fitting. Suppose each user has his or her own utility function that describes the relation between the allocated bandwidth and the resultant benefit to that user. Following the standard assumptions, (1) the function depends on the bandwidth of the user, and on the bandwidth of other users only through their total bandwidth, and (2) the function satisfies certain continuity and concavity properties; we show the unique existence and the fairness with some appropriate sense, of the NEP in the allocation game. The contributions of our work are twofold. First, a novel concept called user-grouping NEP is proposed. This concept is corresponding to the new introduction of the group constraint, under which the uniqueness of NEP proved in [15, 17] no longer holds. Based on this concept we give a new definition for the equilibrium point and prove its uniqueness under our assumptions. The fairness of the allocation based on the user-grouping NEP is also proved. Second, we show that the Gauss-Seidel type algorithm in [15, 18] can be modified to yield a sequence converging to the user-grouping NEP. Since the bandwidth allocation is of central concern in networking systems, our results might result in the reinvestigation and reformulation of other networking issues such that more practical approaches can be developed.

2. Preliminaries

Suppose a networking system has 𝑛 users and they compete for the system bandwidth. Each user is assigned with the bandwidth subject to predecided upper and lower bounds. In addition, 𝑚 of the 𝑛 users form a group and the total bandwidth of the 𝑚 users is also subject to a predecided constraint. We would like to design the system bandwidth allocation policy that optimizes the performance index of each user in the game-theoretical sense. For convenience, we use the list of nomenclature shown at the end of the paper.

Assume the utility function for each user depends on the bandwidth of that user and the total bandwidth of other users. That is, the utility function for user 𝑖 in 𝒩 can be written as 𝑈_𝑖(𝑟_𝑖,𝑅). Also, assume the utility function satisfies the following continuity and concavity properties [18]:

Assumption 1. For each utility function 𝑈_𝑖(𝑟_𝑖,𝑅)(a)𝑈_𝑖(𝑟_𝑖,𝑅) is continuously differentiable with respect to 𝑟_𝑖; (b)(𝜕/𝜕𝑟_𝑖)𝑈_𝑖(𝑟_𝑖,𝑅) is strictly decreasing with respect to 𝑟_𝑖 and nonincreasing with respect to 𝑅.

Now we define the allocation function. For user 𝑖 in 𝒩⧵ℳ with the available bandwidth 𝑇_𝑖, the allocation function 𝐴_𝑖 is defined as 𝐴_𝑖𝑇_𝑖=argmax_{𝑟𝑖≤𝑟≤𝑟𝑖}𝑈_𝑖𝑟,𝑟+𝑇−𝑇_𝑖. (1) For user 𝑖 in ℳ with the available bandwidth 𝑇_𝑖 and inside-group information 𝑇^𝑔_𝑖, the allocation function 𝒜_𝑖 is defined as 𝒜_𝑖𝑇_𝑖,𝑇^𝑔_𝑖=argmax_{𝑟𝑖≤𝑟≤𝑟𝑖,𝑅𝑔≤𝑇𝑔𝑖+𝑟≤𝑅𝑔}𝑈_𝑖𝑟,𝑟+𝑇−𝑇_𝑖. (2)

Assumption 2. The bandwidth allocation with the constraint parameters has the following properties: (a)𝑇_𝑖>𝒜_𝑖(𝑇_𝑖,𝑇^𝑔_𝑖) for each feasible 𝑇_𝑖 and 𝑇^𝑔_𝑖 where 𝑖∈ℳ, and 𝑇_𝑖>𝐴_𝑖(𝑇_𝑖) for each feasible 𝑇_𝑖 where 𝑖∈𝒩⧵ℳ; (b)∑^𝑚_𝑖=1𝑟_𝑖>𝑅_𝑔>𝑅_𝑔>∑^𝑚_𝑖=1𝑟_𝑖.

In our framework, the classical Nash equilibrium point (NEP) in the allocation game is defined as 𝐫^∗∶=𝑟^∗₁,𝑟^∗₂,…,𝑟^∗_𝑛, (3) where 𝑟^∗_𝑖∈argmax_{𝑟∈𝐶𝑖}𝑈_𝑖𝑟,𝑟+_𝑗≠𝑖𝑟^∗_𝑗. (4) Note that 𝐶_𝑖 is {𝑟∣𝑟_𝑖≤𝑟≤𝑟_𝑖,𝑅_𝑔≤𝑟+∑_{𝑗∈ℳ⧵{𝑖}}𝑟^∗_𝑗≤𝑅_𝑔} if 𝑖∈ℳ and is {𝑟∣𝑟_𝑖≤𝑟≤𝑟_𝑖} otherwise. The user-grouping NEP is defined as 𝐫^∗_𝑔𝑅∶=^∗𝑔𝑚,𝑟^∗_𝑚+1,𝑟^∗_𝑚+2,…,𝑟^∗_𝑛, (5) where 𝑅^∗_𝑔∶=^𝑚_𝑖=1𝑟^∗_𝑖. (6)

Remark 3. The definitions (3)-(4) reflect the central concept of the well-studied constrained NEP. That is, given the constrained strategy space 𝐶_𝑖 for each user 𝑖∈𝒩, 𝑟^∗_𝑖 is defined as the maximizer of the utility function of user 𝑖 provided that the strategy 𝑟^∗_𝑗 is adopted by user 𝑗 for each 𝑗∈𝒩⧵{𝑖}. Note that the bandwidth of users in the group ℳ should satisfy the extra group constraint and thus 𝑟^∗_𝑖 might lose its uniqueness in 𝐶_𝑖 as the group constraint is active. The novel concept of user-grouping NEP in (5)-(6) is thus proposed to compensate the property of the equilibrium point. We will show in Section 3.1 that our setting ensures the uniqueness of the user-grouping NEP.

Remark 4. Suppose 𝑇^∗_𝑖∶=𝑇−∑^𝑛_𝑗≠𝑖𝑟^∗_𝑗. Also, let 𝑇_𝑖^𝑔∗∶=𝑅_𝑔−∑^𝑚_𝑗≠𝑖𝑟^∗_𝑗, then 𝑟^∗_𝑖𝒜∶=^𝑖𝑇^∗_𝑖,𝑇_𝑖^𝑔∗𝐴if𝑖∈ℳ_𝑖𝑇^∗_𝑖if𝑖∈𝒩⧵ℳ. (7) By part (a) in Assumption 2 we have 𝑟^∗_𝑖<𝑇^∗_𝑖=𝑇−^𝑛_𝑗≠𝑖𝑟^∗_𝑗. (8) Consequently, 𝑅^∗∶=^𝑛_𝑖=1𝑟^∗_𝑖<𝑇. (9) This means that the NEP, or 𝐫^∗, satisfies the natural constraint 𝑅^∗≤𝑇 and the constraint is always inactive. Part (b) in Assumption 2 is a natural condition such that the constraint 𝑅_𝑔≤∑^𝑚_𝑖=1𝑟_𝑖≤𝑅_𝑔 makes sense.

The existence of the NEP in our setting is guaranteed by Rosen’s result in the following.

Theorem 5 (see [22, Theorem 1]). An equilibrium point exists for every concave 𝑛-person game.

Theorem 5 can be obtained using the classical Kakutani fixed point theorem and in some sense generalizes Nash’s setting on the strategy space of the users [23, 24]. In the next section we delve into other properties and propose an algorithm to locate the NEP.

3. Main Results

3.1. Uniqueness

Our first result is concerned with the uniqueness of the user-grouping NEP. This property as shown in [18, page 13] is not implied by the uniqueness theorem in [22]. For the NEP 𝐫^∗=(𝑟^∗₁,𝑟^∗₂,…,𝑟^∗_𝑛) defined in (3)-(4), the Karash-Kuhn-Tucker (KKT) conditions must be satisfied. That is, for each 𝑖∈𝒩⧵ℳ there exist KKT multipliers 𝜆^∗_𝑖 and 𝜆^∗_𝑖 (see e.g., [25, page 458]) such that −𝜕𝜕𝑟_𝑖𝑈_𝑖𝑟^∗_𝑖,𝑅^∗+𝜆^∗_𝑖−𝜆^∗_𝑖=0,𝜆^∗_𝑖𝑟^∗_𝑖−𝑟_𝑖𝜆=0,_∗𝑖𝑟_𝑖−𝑟^∗_𝑖𝑟=0,_𝑖≤𝑟^∗_𝑖≤𝑟_𝑖,𝜆^∗_𝑖≥0,𝜆^∗_𝑖≥0. (10)(11)(12)(13)(14) In addition, for each 𝑖∈ℳ there exist KKT multipliers 𝜆^∗_𝑖 and 𝜆^∗_𝑖 satisfying (11)–(14), and 𝛾^∗_𝑖 and 𝛾^∗_𝑖 satisfying −𝜕𝜕𝑟_𝑖𝑈_𝑖𝑟^∗_𝑖,𝑅^∗+𝜆^∗_𝑖−𝜆^∗_𝑖+𝛾^∗_𝑖−𝛾^∗_𝑖=0,𝛾^∗_𝑖𝑅^∗_𝑔−𝑅_𝑔𝛾=0,_∗𝑖𝑅_𝑔−𝑅^∗_𝑔𝑅=0,_𝑔≤𝑅^∗_𝑔≤𝑅_𝑔,𝛾^∗_𝑖≥0,𝛾^∗_𝑖≥0, (15)(16)(17)(18)(19) where 𝑅^∗_𝑔 is defined in (6).

Lemma 6. For each 𝑖∈𝒩⧵ℳ, let 𝑟_𝑖⁽¹⁾∶=𝐴_𝑖(𝑇_𝑖⁽¹⁾) and 𝑟_𝑖⁽²⁾∶=𝐴_𝑖(𝑇_𝑖⁽²⁾) for some feasible 𝑇_𝑖⁽¹⁾ and 𝑇_𝑖⁽²⁾, then 𝑟_𝑖⁽¹⁾>𝑟_𝑖⁽²⁾⇒𝜆_𝑖⁽¹⁾−𝜆_𝑖⁽¹⁾≥𝜆_𝑖⁽²⁾−𝜆_𝑖⁽²⁾, (20) where the nonnegative KKT multipliers 𝜆_𝑖⁽¹⁾, 𝜆_𝑖⁽¹⁾,  𝜆_𝑖⁽²⁾ and 𝜆_𝑖⁽²⁾ satisfy 𝜆_𝑖⁽¹⁾𝑟_𝑖⁽¹⁾−𝑟_𝑖=0,𝜆_𝑖⁽¹⁾𝑟_𝑖−𝑟_𝑖⁽¹⁾=0,𝜆_𝑖⁽²⁾𝑟_𝑖⁽²⁾−𝑟_𝑖=0,𝜆_𝑖⁽²⁾𝑟_𝑖−𝑟_𝑖⁽²⁾=0. (21)(22)

Proof. Since 𝑟_𝑖⁽¹⁾ and 𝑟_𝑖⁽²⁾ are both feasible, 𝑟_𝑖⁽¹⁾>𝑟_𝑖⁽²⁾ implies 𝜆_𝑖⁽¹⁾=0 and 𝜆_𝑖⁽²⁾=0 by (21) and (22), respectively. Consequently, the result follows since 𝜆_𝑖⁽¹⁾≥0 and 𝜆_𝑖⁽²⁾≥0.

Theorem 7. The user-grouping NEP defined in (5) is unique.

Proof. Suppose 𝐫⁽¹⁾=(𝑟₁⁽¹⁾,𝑟₂⁽¹⁾,…,𝑟_𝑛⁽¹⁾) and 𝐫⁽²⁾=(𝑟₁⁽²⁾,𝑟₂⁽²⁾,…,𝑟_𝑛⁽²⁾) are both the equilibrium points. Let 𝑅⁽¹⁾=∑^𝑛_𝑖=1𝑟_𝑖⁽¹⁾ and 𝑅⁽²⁾=∑^𝑛_𝑖=1𝑟_𝑖⁽²⁾. We can thus write for 𝑖∈𝒩⧵ℳ that −𝜕𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖⁽¹⁾,𝑅⁽¹⁾+𝜆_𝑖⁽¹⁾−𝜆_𝑖⁽¹⁾−𝜕=0,𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖⁽²⁾,𝑅⁽²⁾+𝜆_𝑖⁽²⁾−𝜆_𝑖⁽²⁾=0, (23) where 𝜆_𝑖⁽¹⁾, 𝜆_𝑖⁽¹⁾,  𝜆_𝑖⁽²⁾, 𝜆_𝑖⁽²⁾ are the associated KKT multipliers. Assume that 𝑅⁽¹⁾>𝑅⁽²⁾. If there exists 𝑖∈𝒩⧵ℳ such that 𝑟_𝑖⁽¹⁾>𝑟_𝑖⁽²⁾, by Lemma 6 and Assumption 1 we have −𝜕𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖⁽¹⁾,𝑅⁽¹⁾+𝜆_𝑖⁽¹⁾−𝜆_𝑖⁽¹⁾𝜕>−𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖⁽²⁾,𝑅⁽²⁾+𝜆_𝑖⁽²⁾−𝜆_𝑖⁽²⁾, (24) which is a contradiction. We thus have 𝑟_𝑖⁽¹⁾≤𝑟_𝑖⁽²⁾ for 𝑖 in 𝒩⧵ℳ. This implies 𝑅_𝑔⁽¹⁾∶=∑^𝑚_𝑖=1𝑟_𝑖⁽¹⁾>∑^𝑚_𝑖=2𝑟_𝑖⁽²⁾=𝑅_𝑔⁽²⁾. Note that, for 𝑖∈ℳ, −𝜕𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖⁽¹⁾,𝑅⁽¹⁾+𝜆_𝑖⁽¹⁾−𝜆_𝑖⁽¹⁾+𝛾_𝑖⁽¹⁾−𝛾_𝑖⁽¹⁾−𝜕=0,𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖⁽²⁾,𝑅⁽²⁾+𝜆_𝑖⁽²⁾−𝜆_𝑖⁽²⁾+𝛾_𝑖⁽²⁾−𝛾_𝑖⁽²⁾=0. (25) With similar arguments in proving Lemma 6 we can show that 𝑅_𝑔⁽¹⁾>𝑅_𝑔⁽²⁾⇒𝛾_𝑖⁽¹⁾−𝛾_𝑖⁽¹⁾≥𝛾_𝑖⁽²⁾−𝛾_𝑖⁽²⁾. (26) If 𝑟_𝑖⁽¹⁾>𝑟_𝑖⁽²⁾ for some 𝑖∈ℳ, we have −𝜕𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖⁽¹⁾,𝑅⁽¹⁾+𝜆_𝑖⁽¹⁾−𝜆_𝑖⁽¹⁾+𝛾_𝑖⁽¹⁾−𝛾_𝑖⁽¹⁾𝜕>−𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖⁽²⁾,𝑅⁽²⁾+𝜆_𝑖⁽²⁾−𝜆_𝑖⁽²⁾+𝛾_𝑖⁽²⁾−𝛾_𝑖⁽²⁾, (27) which is a contradiction. We thus have 𝑟_𝑖^(𝑖)≤𝑟_𝑖⁽²⁾ for each 𝑖∈ℳ and therefore 𝑅_𝑔⁽¹⁾≤𝑅_𝑔⁽²⁾, also a contradiction. With analogous arguments we can show that assuming 𝑅⁽¹⁾<𝑅⁽²⁾ also leads to a contradiction. Therefore 𝑅⁽¹⁾=𝑅⁽²⁾ and by Lemma 6  𝑟_𝑖⁽¹⁾=𝑟_𝑖⁽²⁾ for 𝑖∈𝒩⧵ℳ. This implies 𝑅_𝑔⁽¹⁾=𝑅_𝑔⁽²⁾; therefore 𝐫^∗_𝑔 is unique.

3.2. Fairness

A bandwidth allocation 𝐫=(𝑟₁,𝑟₂,…,𝑟_𝑛) is said to be fair if for any feasible 𝑡₁ and 𝑡₂𝒜_𝑖𝑡₁,𝑡₂≥𝒜_𝑗𝑡₁,𝑡₂⇒𝑟_𝑖≥𝑟_𝑗, (28) where 𝑖,𝑗∈ℳ, and for any feasible 𝑡𝐴_𝑖(𝑡)≥𝐴_𝑗(𝑡)⇒𝑟_𝑖≥𝑟_𝑗, (29) where 𝑖,𝑗∈𝒩⧵ℳ. This definition suggests that a fair allocation guarantees the user in greater need of bandwidth actually obtains more bandwidth.

Theorem 8. The bandwidth allocation based on the NEP defined in (3)-(4) is fair.

Proof. Suppose 𝑖∈ℳ. Let 𝑟_𝑖⁽¹⁾=𝒜_𝑖(𝑇_𝑖⁽¹⁾,𝑡) and 𝑟_𝑖⁽²⁾=𝒜_𝑖(𝑇_𝑖⁽¹⁾+Δ𝑇,𝑡+Δ𝑇). We thus have −𝜕𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖⁽¹⁾,𝑅⁽¹⁾+𝜆_𝑖⁽¹⁾−𝜆_𝑖⁽¹⁾+𝛾_𝑖⁽¹⁾−𝛾_𝑖⁽¹⁾−𝜕=0,𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖⁽²⁾,𝑅⁽²⁾+𝜆_𝑖⁽²⁾−𝜆_𝑖⁽²⁾+𝛾_𝑖⁽²⁾−𝛾_𝑖⁽²⁾=0. (30) Assume 𝑟_𝑖⁽²⁾−Δ𝑇>𝑟_𝑖⁽¹⁾, which implies 𝑟_𝑖⁽²⁾>𝑟_𝑖⁽¹⁾ and by Lemma 6  𝜆_𝑖⁽²⁾−𝜆_𝑖⁽²⁾≥𝜆_𝑖⁽¹⁾−𝜆_𝑖⁽¹⁾. Also, 𝑅⁽²⁾∶=𝑟_𝑖⁽²⁾+𝑅⁽²⁾_−𝑖>𝑟_𝑖⁽¹⁾+Δ𝑇+𝑅⁽²⁾_−𝑖=𝑟_𝑖⁽¹⁾+𝑅⁽¹⁾_−𝑖∶=𝑅⁽¹⁾. (31) Similarly, 𝑅_𝑔⁽²⁾∶=𝑟_𝑖⁽²⁾+^𝑚_𝑗≠𝑖𝑟_𝑗⁽²⁾>𝑟_𝑖⁽¹⁾+Δ𝑇+^𝑚_𝑗≠𝑖𝑟_𝑗⁽²⁾=𝑟_𝑖⁽¹⁾+^𝑚_𝑗≠𝑖𝑟_𝑗⁽¹⁾∶=𝑅_𝑔⁽¹⁾. (32) Note that (32) implies 𝛾_𝑖⁽²⁾−𝛾_𝑖⁽²⁾≥𝛾_𝑖⁽¹⁾−𝛾_𝑖⁽¹⁾. We then have −𝜕𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖⁽²⁾,𝑟_𝑖⁽²⁾+𝑅⁽²⁾_−𝑖+𝜆_𝑖⁽²⁾−𝜆_𝑖⁽²⁾+𝛾_𝑖⁽²⁾−𝛾_𝑖⁽²⁾𝜕>−𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖⁽¹⁾,𝑟_𝑖⁽¹⁾+𝑅⁽¹⁾_−𝑖+𝜆_𝑖⁽¹⁾−𝜆_𝑖⁽¹⁾+𝛾_𝑖⁽¹⁾−𝛾_𝑖⁽¹⁾, (33) which is a contradiction. Therefore, 𝑟_𝑖⁽²⁾−Δ𝑇≤𝑟_𝑖⁽¹⁾, which implies 𝑇_𝑖⁽¹⁾+Δ𝑇−𝑟_𝑖⁽²⁾≥𝑇_𝑖⁽¹⁾−𝑟_𝑖⁽¹⁾, (34) namely, 𝑇_𝑖⁽¹⁾+Δ𝑇−𝒜_𝑖𝑇_𝑖⁽¹⁾+Δ𝑇,𝑡+Δ𝑇≥𝑇_𝑖⁽¹⁾−𝒜_𝑖𝑇_𝑖⁽¹⁾,𝑡. (35) To show the allocation based on 𝐫^∗ is fair, consider first the case that 𝑖,𝑗∈ℳ. By definition 𝑟^∗_𝑖=𝒜_𝑖(𝑇^∗_𝑖,𝑇_𝑖^𝑔∗), 𝑟^∗_𝑗∶=𝒜_𝑗(𝑇^∗_𝑗,𝑇_𝑗^𝑔∗) where 𝑇^∗_𝑖, 𝑇_𝑖^𝑔∗, 𝑇^∗_𝑗 and 𝑇_𝑗^𝑔∗ are all feasible. With 𝑅^∗ defined in (9) we have 𝑇−𝑅^∗=𝑇^∗_𝑖−𝒜_𝑖𝑇^∗_𝑖,𝑇_𝑖^𝑔∗=𝑇^∗_𝑗−𝒜_𝑗𝑇^∗_𝑗,𝑇_𝑖^𝑔∗, (36) which equals the remaining bandwidth of the system after allocation. Given 𝒜_𝑖(𝑇^∗_𝑖,𝑇_𝑖^𝑔∗)>𝒜_𝑗(𝑇^∗_𝑖,𝑇_𝑖^𝑔∗), we have 𝑇^∗_𝑖−𝒜_𝑗𝑇^∗_𝑖,𝑇_𝑖^𝑔∗>𝑇^∗_𝑖−𝒜_𝑖𝑇^∗_𝑖,𝑇_𝑖^𝑔∗=𝑇^∗_𝑗−𝒜_𝑗𝑇^∗_𝑗,𝑇_𝑗^𝑔∗. (37) Note that 𝑇^∗_𝑖−𝑇^∗_𝑗=𝑇_𝑖^𝑔∗−𝑇_𝑗^𝑔∗=𝑟^∗_𝑖−𝑟^∗_𝑗∶=Δ𝑇^∗. (38) Equations (35) and (37) imply Δ𝑇^∗>0; hence 𝑟^∗_𝑖>𝑟^∗_𝑗. The case for 𝑖,𝑗∈𝒩⧵ℳ can be similarly proved and is thus ignored (see [18, Theorem 2.3] for details).

3.3. Algorithm

In this section we analyze the scheme to identify the user-grouping Nash equilibrium point. We say an individual update is implemented on user 𝑖 if the bandwidth of each user other than 𝑖 is fixed and the bandwidth of user 𝑖 is updated to maximize his or her utility function. In addition, we say a batch update occurs in the collection 𝒦 of users if the bandwidth of each user not in 𝒦 is fixed and the individual update is sequentially implemented on each user in 𝒦 repeatedly till an equilibrium is reached. Here 𝒦 is either ℳ or 𝒩⧵ℳ. If 𝒦=ℳ the batch update is implemented assuming no group constraint, namely, 𝑅_𝑔=∞ and 𝑅_𝑔=0. Note that the batch update is guaranteed to reach an equilibrium (see [15] and [18, Section 2.4]). As a result, suppose 𝐫^𝑘=(𝑟^𝑘₁,𝑟^𝑘₁,…,𝑟^𝑘_𝑛) is the system allocation at step 𝑘. If an individual update occurs at user 𝑖∈𝒩⧵ℳ at step 𝑘+1, that means 𝑟_𝑗^𝑘+1=𝑟^𝑘_𝑗 for 𝑗∈𝒩⧵{𝑖}, and −𝜕𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖^𝑘+1,𝑅^𝑘+1+𝜆_𝑖^𝑘+1−𝜆_𝑖^𝑘+1=0, (39) for some KKT multipliers 𝜆_𝑖^𝑘+1 and 𝜆_𝑖^𝑘+1 and 𝑅^𝑘+1=∑^𝑛_𝑖=1𝑟_𝑖^𝑘+1. If a batch update occurs at the group of users at step 𝑘+1, that means 𝑟_𝑗^𝑘+1=𝑟^𝑘_𝑗 for each 𝑗∈𝒩⧵ℳ, and we can write for each 𝑖∈ℳ−𝜕𝜕𝑟_𝑖𝑈_𝑖𝑟_𝑖^𝑘+1,𝑅^𝑘+1+𝜆_𝑖^𝑘+1−𝜆_𝑖^𝑘+1+𝛾_𝑖^𝑘+1−𝛾_𝑖^𝑘+1=0, (40) where 𝜆_𝑖^𝑘+1, 𝜆_𝑖^𝑘+1, 𝛾_𝑖^𝑘+1 and 𝛾_𝑖^𝑘+1 are the associated KKT multipliers. We now show that a repeated implementation of sequential batch updates on users in ℳ and users in 𝒩⧵ℳ, as outline in Algorithm 1, yields a sequence converging to the user-grouping NEP 𝐫^∗_𝑔 in (5). Define first the error measure 𝑒(𝐫^𝑘_𝑔) between 𝐫^𝑘_𝑔 and 𝐫^∗_𝑔 as 𝑒𝐫^𝑘_𝑔=^𝑛_𝑖=𝑚+1𝑟^𝑘_𝑖−𝑟^∗_𝑖+𝑅^𝑘_𝑔−𝑅^∗_𝑔+𝑅^𝑘−𝑅^∗, (41) where 𝑅^𝑘_𝑔∶=∑^𝑚_𝑖=1𝑟^𝑘_𝑖. 𝑅^∗_𝑔 and 𝑅^∗ are defined in (6) and (9), respectively.

(P1) Initiate the algorithm with a feasible 𝐫𝑔 and set 𝑠=0. (P2) (if current step is 𝑘) Perform a batch update for users in ℳ at step 𝑘+1   by assigning 𝑟𝑗𝑘+1=𝑟𝑘𝑗  for all 𝑗∈𝒩⧵ℳ,   to reach 𝑟𝑖𝑘+1=𝒜𝑖𝑇𝑖𝑘+1,𝑇𝑖𝑔,𝑘+1  for all 𝑖∈ℳ. (P3) (if current step is 𝑘) Perform an individual update at user 𝑖 in 𝒩⧵ℳ   by assigning 𝑟𝑘𝑗+1=𝑟𝑘𝑗  for all 𝑗∈𝒩⧵{𝑖},   to reach 𝑟𝑘𝑖+1=𝐴𝑖(𝑇𝑘𝑖), and let 𝑘=𝑘+1.   Sequentially implement this whole procedure till each user in 𝒩⧵ℳ   is updated. (P4) Repeat (P3) to complete a batch update at users in 𝒩⧵ℳ. (P5) Set 𝑠=𝑠+1 and record the current 𝐫𝑔 as ℛ(𝑠).   If ‖ℛ(𝑠)−ℛ(𝑠−1)‖<𝜀 then stops, otherwise go to (P2).

Algorithm 1 

The scheme to locate the user-grouping NEP.

Lemma 9. The error measure 𝑒(𝐫^𝑘_𝑔) defined in (41) is nonincreasing, namely, 𝑒(𝐫_𝑔^𝑘+1)≤𝑒(𝐫^𝑘_𝑔) for any positive integer 𝑘.

Proof. If at step 𝑘+1 an individual update occurs at 𝑖∈𝒩⧵ℳ, (39) is satisfied. Since (10) is also satisfied, we have by Lemma 6 and part (b) in Assumption 1 that 𝑅^𝑘+1≥𝑅^∗⇒𝑟_𝑖^𝑘+1≤𝑟^∗_𝑖,𝑅^𝑘+1≤𝑅^∗⇒𝑟_𝑖^𝑘+1≥𝑟^∗_𝑖. (42) Under the assumption 𝑒𝐫_𝑔^𝑘+1−𝑒𝐫^𝑘_𝑔=𝑟_𝑖^𝑘+1−𝑟^∗−𝑟^𝑘_𝑖−𝑟^∗+𝑅^𝑘+1−𝑅^∗−𝑅^𝑘−𝑅^∗. (43) Suppose 𝑅^𝑘+1≥𝑅^∗. If 𝑅^𝑘≥𝑅^𝑘+1, then 𝑒𝐫_𝑔^𝑘+1−𝑒𝐫^𝑘_𝑔=−𝑟_𝑖^𝑘+1−𝑟^∗−𝑟^𝑘_𝑖−𝑟^∗+𝑅^𝑘+1−𝑅^𝑘⇒=0if𝑟^𝑘_𝑖<𝑟^∗_𝑖<0o.w. (44) Since 𝑅^𝑘<𝑅^𝑘+1 implies 𝑟^𝑘_𝑖<𝑟_𝑖^𝑘+1, we have 𝑒𝐫_𝑔^𝑘+1−𝑒𝐫^𝑘_𝑔=−𝑟_𝑖^𝑘+1+𝑟^𝑘_𝑖+𝑅^𝑘+1−𝑅^∗−𝑅^𝑘−𝑅^∗⇒=0if𝑅^𝑘>𝑅^∗<0o.w. (45) Using similar arguments we can analyze the case for 𝑅^𝑘+1<𝑅^∗ and obtain also that 𝑒(𝐫_𝑔^𝑘+1)≤𝑒(𝐫^𝑘_𝑔). If at step 𝑘+1 a batch update occurs, (15) and (40) hold for each 𝑖∈ℳ. Suppose 𝑅^𝑘+1≥𝑅^∗. If there exists an 𝑖∈ℳ such that 𝑟_𝑖^𝑘+1>𝑟^∗_𝑖 then part (b) in Assumption 1 implies 𝛾_𝑖^𝑘+1−𝛾_𝑖^𝑘+1<𝛾^∗_𝑖−𝛾^∗_𝑖; (46) therefore by (26) 𝑅^∗_𝑔≥𝑅_𝑔^𝑘+1. If no such 𝑖 exists, namely, 𝑟_𝑖^𝑘+1≤𝑟^∗ for each 𝑖∈ℳ, then naturally 𝑅_𝑔^𝑘+1≤𝑅^∗_𝑔. A similar result can be derived for the case 𝑅^𝑘+1≤𝑅^∗. We then have 𝑅^𝑘+1≥𝑅^∗⇒𝑅_𝑔^𝑘+1≤𝑅^∗_𝑔,𝑅^𝑘+1≤𝑅^∗⇒𝑅^𝑘+1≥𝑅^∗_𝑔. (47)(48) Now 𝑒𝐫_𝑔^𝑘+1−𝑒𝐫^𝑘_𝑔=𝑅_𝑔^𝑘+1−𝑅^∗_𝑔−𝑅^𝑘_𝑔−𝑅^∗_𝑔+𝑅^𝑘+1−𝑅^∗−𝑅^𝑘−𝑅^∗. (49) Assume 𝑅^𝑘+1≥𝑅^∗. If 𝑅^𝑘≥𝑅^𝑘+1 then 𝑒𝐫_𝑔^𝑘+1−𝑒𝐫^𝑘_𝑔=−𝑅_𝑔^𝑘+1−𝑅^∗_𝑔−𝑅^𝑘_𝑔−𝑅^∗_𝑔+𝑅^𝑘+1−𝑅^𝑘⇒=0if𝑅^𝑘_𝑔<𝑅^∗_𝑔<0o.w. (50) If 𝑅^𝑘<𝑅^𝑘+1, which is equivalent to 𝑅^𝑘_𝑔<𝑅_𝑔^𝑘+1, then 𝑒𝐫_𝑔^𝑘+1−𝑒𝐫^𝑘_𝑔=−𝑅_𝑔^𝑘+1+𝑅^𝑘_𝑔+𝑅^𝑘+1−𝑅^∗−𝑅^𝑘−𝑅^∗⇒=0if𝑅^𝑘>𝑅^∗<0o.w. (51) Applying similar arguments again we can analyze the case for 𝑅^𝑘+1<𝑅^∗ and obtain also that 𝑒(𝐫_𝑔^𝑘+1)≤𝑒(𝐫^𝑘_𝑔).

Theorem 10. Algorithm 1 yields a sequence {𝐫^𝑘_𝑔}^∞_𝑘=1 converging to 𝐫^∗_𝑔, the user-grouping NEP of the bandwidth allocation game.

Proof. In the light of Lemma 9, we only need to show that for each positive integer 𝑘, 𝐫^𝑘_𝑔≠𝐫^∗_𝑔 implies the existence of a finite integer 𝑘₁ such that 𝑒(𝐫^𝑘+𝑘1_𝑔)<𝑒(𝐫^𝑘_𝑔). Suppose it is not the case Then there exists some 𝑘 such that 𝑒(𝐫^𝑘+𝑘2_𝑔)=𝑒(𝐫^𝑘_𝑔) for any positive integer 𝑘₂, where 𝐫^𝑘_𝑔≠𝐫^∗_𝑔. Consider the update scheme that, at step 𝑘+𝑖−𝑚 for 𝑖∈𝒩⧵ℳ, the individual update occurs at user 𝑖, and at step 𝑘+𝑛+1−𝑚 the batch update takes place for users in ℳ. Without loss of generality we assume 𝑅^{𝑘+𝑛+1−𝑚}≥𝑅^∗, then 𝑅_𝑔(𝑘+𝑛+1−𝑚)≤𝑅^∗_𝑔 by (47). Since 𝑒(𝐫_𝑔^{𝑘+𝑛+1−𝑚})=𝑒(𝐫_𝑔^{𝑘+𝑛−𝑚}), (50) together with (51) implies 𝑅^{𝑘+𝑛−𝑚}≥𝑅^∗; hence 𝑅_𝑔^{𝑘+𝑛−𝑚}≤𝑅^∗_𝑔. That is, 𝑅^⋅−𝑅^∗ and 𝑅^⋅_𝑔−𝑅^∗_𝑔 do not change their signs as the step number is increased from 𝑘+𝑛−𝑚 to 𝑘+𝑛+1−𝑚. Moreover, 𝑅^{𝑘+𝑛−𝑚}≥𝑅^∗ implies 𝑟_𝑛^{𝑘+𝑛−𝑚}≤𝑟^∗_𝑛. Since 𝑒(𝐫_𝑔^{𝑘+𝑛+1−𝑚})=𝑒(𝐫_𝑔^{𝑘+𝑖−𝑚−1}) for 𝑖∈𝒩⧵ℳ, (44) and (45) together imply 𝑅^{𝑘+𝑖−𝑚}≥𝑅^∗ and thus 𝑟_𝑖^{𝑘+𝑖−𝑚}≤𝑟^∗_𝑖, for 𝑖∈𝒩⧵ℳ. As a result, 𝑟_𝑖^{𝑘+𝑛−𝑚}≤𝑟^∗_𝑖 for 𝑖∈𝒩⧵ℳ. Note that 𝑅_𝑔^{𝑘+𝑛−𝑚}≤𝑅^∗_𝑔 and 𝑅^{𝑘+𝑛−𝑚}=𝑅_𝑔^{𝑘+𝑛−𝑚}+^𝑛_𝑖=𝑚+1𝑟_𝑖^{𝑘+𝑛−𝑚}≥𝑅^∗=𝑅^∗_𝑔+^𝑛_𝑖=𝑚+1𝑟^∗_𝑖. (52) We conclude that 𝑅_𝑔^{𝑘+𝑛−𝑚}=𝑅^∗_𝑔 and 𝑟_𝑖^{𝑘+𝑛−𝑚}=𝑟^∗_𝑖 for 𝑖∈𝒩⧵ℳ, namely, 𝐫^𝑘_𝑔=𝐫^∗_𝑔, a contradiction.

4. A Numerical Example

Consider a data communication network system with 100 users. Suppose 30 of them form a group. We then have 𝒩={1,2,…,100} and ℳ={1,2,…,30}. The adopted utility function is 𝑈_𝑖𝑟_𝑖,𝑅=𝑟^𝛼𝑖_𝑖(𝑇−𝑅), (53) where the parameters 𝛼_𝑖’s are listed in Table 1. Note that the utility function, known as the generalized power function [26], has the continuity and concavity properties required by Assumption 1. In particular, it can be shown [18, page 11] easily that the maximizer argmax_𝑟𝑈_𝑖()=𝛼𝑟,𝑅^𝑖1+𝛼_𝑖𝑇_𝑖<𝑇_𝑖, (54) and thus part (a) in Assumption 2 is satisfied. Suppose the total available bandwidth 𝑇=6000, and the upper and lower bounds for total bandwidth allocated to the group is 𝑅_𝑔=2000 and 𝑅_𝑔=800, respectively. Assume that the individual bandwidth constraint for each user in Table 2 is used. Clearly these parameters satisfy the natural requirements of part (b) in Assumption 2. Applying Algorithm 1 yields a dynamic bandwidth allocation evolving with the implementation step, as shown in Figure 2. The left part of the figure shows the evolution of total bandwidth allocated to the group, which is composed of user 1, user 2, up to user 30. At the beginning of the algorithm, an initial feasible bandwidth allocation is allocated to each user of the system. Fix the total bandwidth allocated to the users not in the group and find the optimal total bandwidth 𝑅_𝑔. In the example 𝑅_𝑔 is 3950.3. Since this value is greater than the upper bound 𝑅_𝑔=2000. 𝑅_𝑔 is replaced with 𝑅_𝑔. Fix this 𝑅_𝑔 we have the total available bandwidth 𝑇−𝑅_𝑔=6000−2000=4000 for users not in the group. Based on this availability of the bandwidth we can find the equilibrium point for users not in the group. In the example we have, for instance, the bandwidth 𝑟₇₀=50, 𝑟₉₀=57.554, and 𝑟₁₀₀=38.0004. Now fix the total bandwidth of the users not in the group, and find the optimal bandwidth 𝑅_𝑔 again. In the example we obtain 𝑅_𝑔=2115.8. Since this value is greater than the upper bound 𝑅_𝑔=2000, 𝑅_𝑔 is replaced with 𝑅_𝑔 again. Note that the current optimal bandwidth allocation for each user outside the group is found based on the condition that the total bandwidth for the group is 𝑅_𝑔. The current 𝑅_𝑔 and 𝑟_𝑚+1,…,𝑟_𝑛 is thus the 𝑅^∗_𝑔 and 𝑟^∗_𝑚+1,…,𝑟^∗_𝑛 for the user-grouping NEP in (5).

Table 1 

The parameters of 𝛼_𝑖 for 𝑖∈𝒩={1,2,…,100}.

Table 2 

The bandwidth constraint for each user.

(a)

(b)

Figure 2 

The sequence generated by Algorithm 1 for the example.

5. Conclusion

We have proposed a novel bandwidth allocation model based on game theory. The consideration of the user-grouping constraint distinguishes this model from the abundant ones concerning similar allocation issues. Suppose each user competes for the system bandwidth resources and is granted with a constrained decision space. In particular, some users are united in one group and the total bandwidth allocated to the group is constrained as well. Given the appropriate constraint parameters and the utility function satisfying mild continuity and concavity conditions for each user, we have shown the unique existence of the user-grouping Nash equilibrium point for the allocation game. In addition, we have shown the fairness, in a proper sense, of the allocation based on this equilibrium point. Finally, we have proposed an iterative algorithm and proved that a sequence converging to the point can be generated by the algorithm. A practical example illustrating a network satisfying our settings has been given to show how the equilibrium point can be located successfully.

Nomenclature

𝒩:	The index set for the users, that is, 𝒩∶={1,2,…,𝑛}
ℳ:	The index set for the users in the group, that is, ℳ∶={1,2,…,𝑚}
𝒩⧵ℳ:	The index set for the users not in the group, that is, 𝒩⧵ℳ∶={𝑚,𝑚+1,…,𝑛}
𝑟𝑖:	The bandwidth allocated to user 𝑖
𝑟𝑖:	Lower bound for 𝑟𝑖
𝑟𝑖:	Upper bound for 𝑟𝑖
𝑅𝑔:	Total bandwidth allocated to the group members, that is, 𝑅𝑔∶=𝑟1+𝑟2+⋯+𝑟𝑚
𝑅𝑔:	Lower bound for 𝑅𝑔
𝑅𝑔:	Upper bound for 𝑅𝑔
𝑇:	Total bandwidth available in the system
𝑅:	Total bandwidth allocated, that is, 𝑅∶=𝑟1+𝑟2+⋯+𝑟𝑛
𝑅−𝑖:	Total bandwidth allocated, excluding to user 𝑖, that is, 𝑅−𝑖∶=𝑟1+⋯+𝑟𝑖−1+𝑟𝑖+1+⋯+𝑟𝑛
𝑇𝑖:	Total bandwidth available for user 𝑖, that is, 𝑇𝑖∶=𝑇−𝑅−𝑖
𝑇𝑔𝑖:	Inside-group information for user 𝑖∈ℳ, that is, 𝑇𝑔𝑖∶=𝑅𝑔−∑𝑚𝑗≠𝑖𝑟𝑗.

Acknowledgment

This research was supported by the National Science Council of Taiwan under Grant NSC 100-2221-E-005-071.