This section presents the LTE/WiFi system infrastructure and introduces the fundamental concepts of NBS, PNBS and ZGBS. Then, a new three-phase bargaining game model is formulated to focus on the coexistence of LTE and WiFi networks. Finally, the main steps of our proposed spectrum allocation algorithm are described.

LTE/WiFi platform and interactive bargaining game

In this study, a heterogeneous network platform is considered to establish the joint LTE/WiFi platform. We consider a coexisting network of one LTE-LAA macro-cell and multiple WiFi micro-cells, which are overlaid on the macro-cell coverage area, and they are operated by WiFi access points (APs), i.e., ({mathbb{W}}=left{{mathcal{W}}_{1},dots ,{mathcal{W}}_{m}right}). Due to the small coverage of the APs, interference among them is neglected. LTE base station (left(mathfrak{B}right)) operates on a licensed spectrum band (left({mathcal{S}}_{mathfrak{B}}right)) , while the ({mathcal{W}}_{1le ile m}) has its unlicensed spectrum band (left({mathcal{S}}_{{mathcal{W}}_{i}}right)). The (mathfrak{B}) and ({mathcal{W}}_{i}) are connected via optical fiber networks. We assume that there are a number of IoT devices, i.e., ({mathbb{D}}=left{{mathcal{D}}_{1},dots ,{mathcal{D}}_{n}right}), which are located in the coverage region of the LTE/WiFi network system. The traffic services of these devices are either served by the (mathfrak{B}) or offloaded to the corresponding AP. It is assumed that the association among (mathfrak{B}), ({mathcal{W}}_{1le ile m}) and ({mathcal{D}}_{1le jle n}) has already been determined. The LTE/WiFi combined network infrastructure is considered as shown in Fig. 1, and Table 1 lists the mathematical notations used in this paper [14].

Fig. 1
figure 1

LTE/WiFi platform architecture. In this study, the LTE/WiFi platform architecture stem consist of multiple smart devices, which is denoted as the set of IoT devices

Table 1 Symbols and parameters for mathematical formulations

Individual IoT devices have been equipped with technology to enable them to connect simultaneously with the LTE and WiFi networks. Therefore, the collaboration between (mathfrak{B}) and (mathcal{W}) is realized by the dual connectivity capability, which enables IoT devices to aggregate the ({mathcal{S}}_{mathfrak{B}}) and ({mathcal{S}}_{mathcal{W}}) spectrum resources. According to a Poisson process, individual IoT devices in the ({mathbb{D}}) generate their communication tasks. In the coverage area of ({mathcal{W}}_{i}), we consider two different device sets. The set ({mathbb{B}}_{{mathcal{W}}_{i}}) consists of devices, which generate BESs and the set of ({mathbb{Q}}_{{mathcal{W}}_{i}}) consists of devices, which generate QPSs. These all devices can contact simultaneously to the (mathfrak{B}) and their corresponding (mathcal{W}) to share the ({mathcal{S}}_{mathfrak{B}}) and ({mathcal{S}}_{{mathcal{W}}_{i}}). If the ({mathcal{D}}_{j}) is in the ({mathcal{W}}_{i})’s area, the totally allocated spectrum resource for the ({mathcal{D}}_{j}) is ({mathcal{A}}_{{mathcal{D}}_{j}}={mathcal{A}}_{{mathcal{D}}_{j}}^{mathfrak{B}}+{mathcal{A}}_{{mathcal{D}}_{j}}^{{mathcal{W}}_{i}}) where ({mathcal{A}}_{{mathcal{D}}_{j}}^{mathfrak{B}}) and ({mathcal{A}}_{{mathcal{D}}_{j}}^{{mathcal{W}}_{i}}) are the ({mathcal{D}}_{j})’s LTE and WiFi spectrum amounts, respectively. The operational timeline is discretized into time slots to make spectrum allocation; it is the same time scale of communication task arrivals, which is a common assumption in the LTE/WiFi system [7].

In the LTE/WiFi system, the ({mathcal{S}}_{mathfrak{B}}) and ({mathcal{S}}_{{mathcal{W}}_{i}}) are limited resources. Therefore, effective spectrum allocation strategies should be considered to improve the system performance. To address the ({mathcal{S}}_{mathfrak{B}}) and ({mathcal{S}}_{{mathcal{W}}_{i}}) sharing problem, we formulate three bargaining games, i.e., ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}), ({mathbb{G}}_{mathfrak{B}}) and ({mathbb{G}}_{{mathcal{W}}_{i}}), at each time period. It is noteworthy that these three bargaining games formulate the (mathfrak{B})(mathcal{W})(mathcal{D}) association in a cooperative manner. In the ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}), the ({mathcal{S}}_{mathfrak{B}}) is divided for the (mathfrak{B}) and ({mathcal{W}}_{1le ile m}) using the idea of NBS where ({mathcal{R}}_{mathfrak{B}}) for the (mathfrak{B}) and ({mathcal{R}}_{{mathcal{W}}_{i}}) for the({mathcal{W}}_{i}). In the ({mathbb{G}}_{mathfrak{B}}), the all devices generated QPSs bargain with each other to share the ({mathcal{R}}_{mathfrak{B}}) according to the PNBS. In the ({mathbb{G}}_{{mathcal{W}}_{i}}), the devices in the ({mathbb{B}}_{{mathcal{W}}_{i}}) and ({mathbb{Q}}_{{mathcal{W}}_{i}}) bargain with each other to distribute the ({mathcal{R}}_{i}) and ({mathcal{S}}_{{mathcal{W}}_{i}}) based on the ZGBS. The ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}), ({mathbb{G}}_{mathfrak{B}}) and ({mathbb{G}}_{{mathcal{W}}_{i}}) are operated sequentially in a step-by-step interactive fashion. Formally, we define the game entities, i.e., ({mathbb{G}}=left{{mathbb{G}}_{mathcal{W}}^{mathfrak{B}}, {mathbb{G}}_{mathfrak{B}}, {mathbb{G}}_{{mathcal{W}}_{i}}right}=left{{mathbb{W}},{mathbb{D}},mathfrak{B}, left{{mathbb{G}}_{mathcal{W}}^{mathfrak{B}}|{mathbb{Q}}_{mathfrak{B}},{mathbb{B}}_{{mathcal{W}}_{i}},{mathcal{W}}_{i}in {mathbb{W}},{mathcal{S}}_{mathfrak{B}},left({mathcal{R}}_{mathfrak{B}} ,{mathcal{R}}_{{mathcal{W}}_{i}}right), {mathfrak{U}}_{mathfrak{B}}left(cdot right),{mathfrak{U}}_{{mathcal{W}}_{i}}left(cdot right)right}, left{{mathbb{G}}_{mathfrak{B}}|{mathbb{Q}}_{mathfrak{B}},{mathcal{R}}_{mathfrak{B}},{Upsilon}_{{mathcal{D}}_{k}in {mathbb{Q}}_{mathfrak{B}}},{U}_{{mathcal{D}}_{k}}left(cdot right)right}, left{{mathbb{G}}_{{mathcal{W}}_{1le ile m}}|{mathcal{W}}_{i},{mathcal{D}}_{j}in left({mathbb{B}}_{{mathcal{W}}_{i}}cup {mathbb{Q}}_{mathfrak{B}}right),{mathcal{R}}_{{mathcal{W}}_{i}},{mathcal{S}}_{{mathcal{W}}_{i}},{mathcal{J}}_{{mathcal{D}}_{j}},{mathcal{U}}_{{mathcal{D}}_{j}}^{QPS}left(cdot right),{mathcal{U}}_{{mathcal{D}}_{j}}^{BES}left(cdot right)right}, Tright}) of gameplay.

  • The ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}), ({mathbb{G}}_{mathfrak{B}}) and ({mathbb{G}}_{{mathcal{W}}_{i}}) are mutually and reciprocally interdependent in an interactive manner, and they work together to share the ({mathcal{S}}_{mathfrak{B}}) and ({mathcal{S}}_{{mathcal{W}}_{i}}).

  • ({mathbb{Q}}_{{mathcal{W}}_{i}}) and ({mathbb{B}}_{{mathcal{W}}_{i}}) are the sets of IoT devices, which are covered by the ({mathcal{W}}_{i}) and generate QPSs and BESs, respectively. ({mathbb{Q}}_{mathfrak{B}}) is the set of all IoT devices, which generate QPSs where ({mathbb{Q}}_{mathfrak{B}}=bigcup_{{mathcal{W}}_{i}in {mathbb{W}}}{mathbb{Q}}_{{mathcal{W}}_{i}}).

  • In the ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}), the (mathfrak{B}) and ({mathcal{W}}_{1le ile m}) are game players, and ({mathcal{R}}_{mathfrak{B}}), ({mathcal{R}}_{{mathcal{W}}_{i}}) are their strategies. For the (mathfrak{B}) and ({mathcal{W}}_{i}), the ({mathfrak{U}}_{mathfrak{B}}left(cdot right)),({mathfrak{U}}_{{mathcal{W}}_{i}}left(cdot right)) are their utility functions, respectively.

  • In the ({mathbb{G}}_{mathfrak{B}}), the device ({mathcal{D}}_{k}in {mathbb{Q}}_{mathfrak{B}}) is a game player, and ({Upsilon}_{{mathcal{D}}_{k}}) is its strategy to share the ({mathcal{R}}_{mathfrak{B}}). ({U}_{{mathcal{D}}_{k}}left(cdot right)) is the ({mathcal{D}}_{k})’s utility function.

  • In the ({mathbb{G}}_{{mathcal{W}}_{i}}), the ({mathcal{D}}_{j}in left{{mathbb{Q}}_{{mathcal{W}}_{i}}cup {mathbb{B}}_{{mathcal{W}}_{i}}right}) is a game player, and ({mathcal{J}}_{{mathcal{D}}_{j}}) is its strategy. If ({mathcal{D}}_{j}in {mathbb{B}}_{{mathcal{W}}_{i}} left({text{ or }} {mathcal{D}}_{j}in {mathbb{Q}}_{{mathcal{W}}_{i}}right)), ({mathcal{U}}_{{mathcal{D}}_{j}}^{BES}left(cdot right) left({text{ or }} {mathcal{U}}_{{mathcal{D}}_{j}}^{QPS}left(cdot right)right)) is its utility function.

  • Discrete time model (Tin left{{t}_{1},dots ,{t}_{c},{t}_{c+1},dots right}) is represented by a sequence of time steps. The length of ({t}_{c}) matches the event time-scale of ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}), ({mathbb{G}}_{mathfrak{B}}) and ({mathbb{G}}_{{mathcal{W}}_{i}}).

The fundamental ideas of PNBS and ZGBS

To characterize the fundamental idea of PNBS, we assume an (n)-player bargaining problem. Let (N=left{1dots idots nright}) be the set of players and ({mathbb{R}}_{+}) (left({text{ or }} {mathbb{R}}_{++}right)) is the set of all non-negative (or all positive) real numbers. Let ({mathbb{R}}_{+}^{n}) (left({text{ or }} {mathbb{R}}_{++}^{n}right)) be the (n)-fold Cartesian product of ({mathbb{R}}_{+}) (left({text{ or }} {mathbb{R}}_{++}right)). For any (x, yin {mathbb{R}}_{+}^{n}), we write (xge y) to mean ({x}_{i}ge {y}_{i}) for all (iin N). (x>y) means ({x}_{i}ge {y}_{i}) for all (iin N) and (xne y). (xgg y) means ({x}_{i}>{y}_{i}) for all (iin N). For any subset (Asubseteq {mathbb{R}}_{+}^{n}), (A) is said to be i) non-trivial if there exists (ain A) such that (agg) 0, and ii) comprehensive if for all (x,yin {mathbb{R}}_{+}^{n}), (x>y) and (xin A) implies (yin A). Let (sum) be the set of all on-trivial, compact and comprehensive subsets of ({mathbb{R}}_{+}^{n}). Elements in (sum) are interpreted as bargaining problems. Usually, a bargaining solution assigns a nonempty subset of (A) for every bargaining problem (Ain sum). For all (Ain sum) and all (iin N), let ({m}_{i}left(Aright)={max}left{{a}_{i}|left({a}_{1},dots {a}_{i}dots {a}_{n}right)in Aright}). Therefore, (mleft(Aright)equiv {left({m}_{i}left(Aright)right)}_{iin N}) is the ideal point of (A). Let ({d}_{iin N}) be a disagree point, which is expected to be the result of non-cooperative actions; it means a failure of the bargaining process. If for all (Ain sum), Nash bargaining solution, i.e., NBS (left(Aright)), is defined by [9]:

$$NBSleft(Aright)=prod_{iin N}left({a}_{i}-{d}_{i}right)ge prod_{iin N}left({x}_{i}-{d}_{i}right), quad {text{s.t.}},; ain A {text{ and }} xin A$$


Based on the NBS (left(Aright)), the PNBS, i.e., PNBS (left(Aright)), is mathematically formulated as follows [9]:

$$PNBSleft(Aright)={{min}}_{iin N}left{frac{{a}_{i}-{d}_{i}}{{m}_{i}left(Aright)-{d}_{i}}right}ge {{min}}_{iin N}left{frac{{x}_{i}-{d}_{i}}{{m}_{i}left(Aright)-{d}_{i}}right}, quad {text{s.t.}},; a,xin NBSleft(Aright)$$


To define the concept of ZGBS, the bargaining problem is represented as a quadruple (left{Gamma , S,u,dright}), where (Gamma) is a partition of a player set (N=left{1dots idots nright}) and each element in (Gamma) represents a formed coalition.(S) is a feasible payoff set in ({mathbb{R}}^{left|Nright|}) where ({mathbb{R}}^{left|Nright|}) is the (n)-dimensional Euclidean space, and (left|Nright|) denotes its size. The (din {mathbb{R}}^{left|Nright|}) is a disagreement payoff vector when the coalitions fail to reach any joint contracts. There exists at least one element (uin S) where ({u}_{i}>{d}_{i}) for any (iin N); ({u}_{i}) represents the (i)’s payoff. When (Gamma) is the finest partition of (N), the pure bargaining problem with a coalition structure is the original Nash’s pure bargaining problem. The ZGBS is a direct extension to Nash’s solution. To express the ZGBS, a function for two payoff vectors (u) and (v) with (Ain Gamma), i.e., (fleft(u,v,Aright)), is defined as [10]:

$$fleft(u,v,Aright)=sum_{Ain Gamma }left(frac{1}{left|Aright|}times left(sum_{iin A}frac{{u}_{i}-{d}_{i}}{{v}_{i}-{d}_{i}}right)right)$$


According to (3), the ZGBS (left(Gamma , S,u,dright)) is defined by [10]:

$$begin{aligned} & ZGBSleft(Gamma , S,uright):= {s}^{*}=underset{uin S}{mathbf{max}}fleft(u,{s}^{*},Aright)=left|Gamma right|\ &quad {text{s.t.}},; uin S, {u}_{i}>{d}_{i} {text{ and }}forall iin Nend{aligned}$$


In 2004, Chae and Heidhues proposed a new solution concept to the group bargaining problem. In their approach, groups of individual players bargain both within and across groups. It treats a bargaining group as one bargainer even in cases where the bargaining group consists of heterogeneous individuals. They also showed that their solution implies the joint-bargaining paradox, and specified a class of solutions in which a larger bargaining group is treated better. The Chae and Heidhues’s solution constitutes a traditional Nash solution within as well as across groups; it is given by [11]:

$$underset{uin S}{mathbf{max}}prod_{j=1}^{k}left(prod_{iin {A}_{j}}left({u}_{i}-{d}_{i} right)right), quad {text{s.t.}},; {A}_{j}in Gamma ={{A}_{1}, …, {A}_{k}}$$


Despite the difference in mathematical representations, the paper [10] has proved that the ZGBS and the Chae and Heidhues’s solution become identical after applying a linear transformation to one problem. Therefore, they are equivalent in group bargaining problems [10].

The three-phase bargaining game in the LTE/WiFi platform

In the LTE/WiFi coexistent platform infrastructure, individual IoT devices generate their tasks. To effectively share the ({mathcal{S}}_{mathfrak{B}}) and ({mathcal{S}}_{mathcal{W}}) spectrum resources, three bargaining games ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}), ({mathbb{G}}_{mathfrak{B}}) and ({mathbb{G}}_{{mathcal{W}}_{i}}) are developed, and work together to allocate the ({mathcal{S}}_{mathfrak{B}}) and ({mathcal{S}}_{mathcal{W}}). At the first phase, the ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}) is designed to partition the ({mathcal{S}}_{mathfrak{B}}) into ({mathcal{R}}_{mathfrak{B}}) and ({mathcal{R}}_{{mathcal{W}}_{1le ile m}}). For this partitioning problem, the (mathfrak{B}) and ({mathcal{W}}_{i}) bargain with each other based on the current information about QPSs and BESs. As game players, the utility functions for (mathfrak{B}) and ({mathcal{W}}_{i}) at time ({t}_{c}), i.e., ({mathfrak{U}}_{mathfrak{B}}^{{t}_{c}}left(cdot right)) and ({mathfrak{U}}_{{mathcal{W}}_{i}}^{{t}_{c}}left(cdot right)), are defined as follows:

$$begin{aligned} & left{begin{array}{c}{mathfrak{U}}_{mathfrak{B}}^{{t}_{c}}left({mathcal{S}}_{mathfrak{B}},{mathcal{R}}_{mathfrak{B}},{mathfrak{M}}_{mathfrak{B}}^{{t}_{c}}right)={log}left(frac{{min}left({mathcal{S}}_{mathfrak{B}},{mathcal{R}}_{mathfrak{B}}right)}{{mathfrak{M}}_{mathfrak{B}}^{{t}_{c}}}+eta right)-expleft(-frac{{min}left({mathcal{S}}_{mathfrak{B}},{mathcal{R}}_{mathfrak{B}}right)}{{mathfrak{M}}_{mathfrak{B}}^{{t}_{c}}}right) \ \ \ {mathfrak{U}}_{{mathcal{W}}_{i}}^{{t}_{c}}left({mathcal{S}}_{mathfrak{B}},{mathcal{S}}_{{mathcal{W}}_{i}},{mathcal{R}}_{{mathcal{W}}_{i}},{mathfrak{M}}_{{mathcal{W}}_{i}}^{{t}_{c}}right)=beta times left(frac{{log}left(frac{{min}left({mathcal{S}}_{mathfrak{B}},{mathcal{R}}_{{mathcal{W}}_{i}}right)+{mathcal{S}}_{{mathcal{W}}_{i}}}{{mathfrak{M}}_{{mathcal{W}}_{i}}^{{t}_{c}}}+kappa right)}{{log}left(frac{{min}left({mathcal{S}}_{mathfrak{B}},{mathcal{R}}_{{mathcal{W}}_{i}}right)+{mathcal{S}}_{{mathcal{W}}_{i}}}{{mathfrak{M}}_{{mathcal{W}}_{i}}^{{t}_{c}}}+omega right)}right) \ end{array}right. \ &qquad {text{s.t.}},; {mathfrak{M}}_{mathfrak{B}}^{{t}_{c}}=sum_{{mathcal{D}}_{k}in {mathbb{Q}}_{mathfrak{B}}}{mathcal{T}}_{{mathcal{D}}_{k}},{mathfrak{M}}_{{mathcal{W}}_{i}}^{{t}_{c}}=sum_{{{mathcal{D}}_{j}in {mathbb{B}}}_{{mathcal{W}}_{i}}}{mathcal{T}}_{{mathcal{D}}_{j}}{text{ and }}{mathcal{S}}_{mathfrak{B}}= {mathcal{R}}_{mathfrak{B}}+ sum_{{mathcal{W}}_{i}in {mathbb{W}}}{mathcal{R}}_{{mathcal{W}}_{i}} \ end{aligned}$$


where (eta) is a control parameter for the ({mathfrak{U}}_{mathfrak{B}}^{{t}_{c}}left(cdot right)), and (beta), (kappa), (omega) are control parameters for the ({mathfrak{U}}_{{mathcal{W}}_{i}}^{{t}_{c}}left(cdot right)). ({mathcal{T}}_{{mathcal{D}}_{k}}) is the ({mathcal{D}}_{k})’s maximum spectrum requirement. For the ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}), the features of efficiency, anonymity and scale invariance are important between QPSs and BESs. Therefore, the idea of classical NBS is adopted. In this study, the utilities of disagreement points are assumed as zeros, and (left[{mathcal{R}}_{mathfrak{B}},{mathcal{R}}_{{mathcal{W}}_{1}},dots ,{mathcal{R}}_{{mathcal{W}}_{m}}right]) is obtained as the NBS for the ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}); it is given by:

$$underset{left[{mathcal{R}}_{mathfrak{B}},{mathcal{R}}_{{mathcal{W}}_{1}},dots ,{mathcal{R}}_{{mathcal{W}}_{m}}right]}{{NBS}}left({mathfrak{U}}_{mathfrak{B}}^{{t}_{c}}left(cdot right),{mathfrak{U}}_{{mathcal{W}}_{1le ile m}}^{{t}_{c}}left(cdot right)right)=left({mathfrak{U}}_{mathfrak{B}}^{{t}_{c}}left({mathcal{R}}_{mathfrak{B}}right)times prod_{{mathcal{W}}_{i}in {mathbb{W}}}{mathfrak{U}}_{{mathcal{W}}_{i}}^{{t}_{c}}left({mathcal{R}}_{{mathcal{W}}_{i}}right)right)ge left({mathfrak{U}}_{mathfrak{B}}^{{t}_{c}}left({mathcal{R}}_{mathfrak{B}}^{^{prime}}right)times prod_{{mathcal{W}}_{i}in {mathbb{W}}}{mathfrak{U}}_{{mathcal{W}}_{i}}^{{t}_{c}}left({mathcal{R}}_{{mathcal{W}}_{i}}^{^{prime}}right)right)$$


At the second phase, the ({mathbb{G}}_{mathfrak{B}}) game is implemented to share the ({mathcal{R}}_{mathfrak{B}}) among devices in the ({mathbb{Q}}_{mathfrak{B}}). For the ({mathcal{D}}_{k}in {mathbb{Q}}_{mathfrak{B}}), the utility function, i.e., ({U}_{{mathcal{D}}_{k}}^{{t}_{c}}left(cdot right)), is defined as follows:

$$begin{aligned} & {U}_{{mathcal{D}}_{k}}^{{t}_{c}}left({Upsilon}_{{mathcal{D}}_{k}},{mathcal{T}}_{{mathcal{D}}_{k}},{mathcal{R}}_{mathfrak{B}}right)=left(frac{{exp}left(frac{{Upsilon}_{{mathcal{D}}_{k}}}{{mathcal{T}}_{{mathcal{D}}_{k}}}right)-{exp}left(-frac{{Upsilon}_{{mathcal{D}}_{k}}}{{mathcal{T}}_{{mathcal{D}}_{k}}}right)}{{exp}left(frac{{Upsilon}_{{mathcal{D}}_{k}}}{{mathcal{T}}_{{mathcal{D}}_{k}}}right)+{exp}left(-frac{{Upsilon}_{{mathcal{D}}_{k}}}{{mathcal{T}}_{{mathcal{D}}_{k}}}right)}right)+frac{{log}left(frac{{Upsilon}_{{mathcal{D}}_{k}}}{{mathcal{T}}_{{mathcal{D}}_{k}}}+psi right)}{{log}left(frac{{Upsilon}_{{mathcal{D}}_{k}}}{{mathcal{T}}_{{mathcal{D}}_{k}}}+varrho right)}\ &qquad {text{s.t.}},; {mathcal{R}}_{mathfrak{B}}ge sum_{{mathcal{D}}_{k}in {mathbb{Q}}_{mathfrak{B}}}{Upsilon}_{{mathcal{D}}_{k}}\ end{aligned}$$


where (psi), (varrho) are control parameters for the ({U}_{{mathcal{D}}_{k}}^{{t}_{c}}left(cdot right)). To get a fair-efficient solution for QPSs, we concern the features of equity principle and weak contraction independence. Therefore, the PNBS is preferred for the solution concept of ({mathbb{G}}_{mathfrak{B}}). As the PNBS for the ({mathbb{G}}_{mathfrak{B}}), (left[dots ,{Upsilon}_{{mathcal{D}}_{k}in {mathbb{Q}}_{mathfrak{B}}},dots right]) is obtained by:

$$begin{aligned} &underset{left[dots ,{Upsilon}_{{mathcal{D}}_{k}in {mathbb{Q}}_{mathfrak{B}}},dots right]}{{PNBS}}left({U}_{{mathcal{D}}_{k}in {mathbb{Q}}_{mathfrak{B}}}^{{t}_{c}}left(cdot right)right)=underset{{mathcal{D}}_{k}}{{min}}left{frac{{U}_{{mathcal{D}}_{k}}^{{t}_{c}}left({Upsilon}_{{mathcal{D}}_{k}}right)}{{m}_{{mathcal{D}}_{k}}left({U}_{{mathcal{D}}_{k}}^{{t}_{c}}left(cdot right)right)}right} ge underset{{mathcal{D}}_{k}}{{min}}left{frac{{U}_{{mathcal{D}}_{k}}^{{t}_{c}}left({Upsilon}_{{mathcal{D}}_{k}}^{^{prime}}right)}{{m}_{{mathcal{D}}_{k}}left({U}_{{mathcal{D}}_{k}}^{{t}_{c}}left(cdot right)right)}right}\ &quad {text{s.t.}},; left[dots ,{Upsilon}_{{mathcal{D}}_{k}in {mathbb{Q}}_{mathfrak{B}}},dots right]in prod_{{mathcal{D}}_{k}in {mathbb{Q}}_{mathfrak{B}}}{U}_{{mathcal{D}}_{k}}^{{t}_{c}}left({Upsilon}_{{mathcal{D}}_{k}}right)ge prod_{{mathcal{D}}_{k}in {mathbb{Q}}_{mathfrak{B}}}{U}_{{mathcal{D}}_{k}}^{{t}_{c}}left({Upsilon}_{{mathcal{D}}_{k}}^{^{prime}}right)\ end{aligned}$$


At the third phase, each ({mathcal{W}}_{i}) distributes its ({mathcal{R}}_{{mathcal{W}}_{i}}) and ({mathcal{S}}_{{mathcal{W}}_{i}}) for its corresponding device ({mathcal{D}}_{j}in left{{mathbb{B}}_{{mathcal{W}}_{i}}cup {mathbb{Q}}_{{mathcal{W}}_{i}}right}). During the ({mathbb{G}}_{{mathcal{W}}_{i}}) game process, the ({mathcal{W}}_{i}) categorizes its corresponding devices into two groups, i.e., ({mathbb{B}}_{{mathcal{W}}_{i}}) and ({mathbb{Q}}_{{mathcal{W}}_{i}}), and the spectrum sharing problem in treated as the group bargaining problem. As a game player, the ({mathcal{D}}_{j})’s utility functions for QPSs and BESs, i.e., ({mathcal{U}}_{{mathcal{D}}_{j}}^{QPS}left(cdot right)) and ({mathcal{U}}_{{mathcal{D}}_{j}}^{BES}left(cdot right)), are defined as follows:

$$begin{aligned} &left{begin{array}{c}{mathcal{U}}_{{mathcal{D}}_{j}}^{QPS}left({mathcal{J}}_{{mathcal{D}}_{j}},{mathcal{S}}_{{mathcal{W}}_{i}},{mathcal{R}}_{{mathcal{W}}_{i}},{mathcal{S}}_{{mathcal{W}}_{i}},{mathbb{B}}_{{mathcal{W}}_{i}},{mathbb{Q}}_{{mathcal{W}}_{i}}right)= \ \ left(frac{expleft(frac{left({mathcal{J}}_{{mathcal{D}}_{j}}+{Upsilon}_{{mathcal{D}}_{j}}right)}{{mathcal{T}}_{{mathcal{D}}_{j}}}right)-expleft(-frac{left({mathcal{J}}_{{mathcal{D}}_{j}}+{Upsilon}_{{mathcal{D}}_{j}}right)}{{mathcal{T}}_{{mathcal{D}}_{j}}}right)}{expleft(frac{left({mathcal{J}}_{{mathcal{D}}_{j}}+{Upsilon}_{{mathcal{D}}_{j}}right)}{{mathcal{T}}_{{mathcal{D}}_{j}}}right)+expleft(-frac{left({mathcal{J}}_{{mathcal{D}}_{j}}+{Upsilon}_{{mathcal{D}}_{j}}right)}{{mathcal{T}}_{{mathcal{D}}_{j}}}right)}right)+frac{{log}left(frac{left({mathcal{J}}_{{mathcal{D}}_{j}}+{Upsilon}_{{mathcal{D}}_{j}}right)}{{mathcal{T}}_{{mathcal{D}}_{j}}}+psi right)}{{log}left(frac{left({mathcal{J}}_{{mathcal{D}}_{j}}+{Upsilon}_{{mathcal{D}}_{j}}right)}{{mathcal{T}}_{{mathcal{D}}_{j}}}+varrho right)}\ \ \ {mathcal{U}}_{{mathcal{D}}_{j}}^{BES}left({mathcal{J}}_{{mathcal{D}}_{j}},{mathcal{S}}_{{mathcal{W}}_{i}},{mathcal{R}}_{{mathcal{W}}_{i}},{mathcal{S}}_{{mathcal{W}}_{i}},{mathbb{B}}_{{mathcal{W}}_{i}},{mathbb{Q}}_{{mathcal{W}}_{i}}right)=frac{alpha }{xi +expleft(-frac{{mathcal{J}}_{{mathcal{D}}_{j}}}{{mathcal{T}}_{{mathcal{D}}_{j}}}right)}-tau end{array}right.\&quad {text{s.t.}},; left({mathcal{R}}_{{mathcal{W}}_{i}}+{mathcal{S}}_{{mathcal{W}}_{i}}right)ge sum_{{mathcal{D}}_{j}in left{{mathbb{B}}_{{mathcal{W}}_{i}}cup {mathbb{Q}}_{{mathcal{W}}_{i}}right}}{mathcal{J}}_{{mathcal{D}}_{j}}\ end{aligned}$$


where (psi), (varrho) are control parameters for the ({mathcal{U}}_{{mathcal{D}}_{j}}^{QPS}left(cdot right)); they are defined as the same in the ({U}_{{mathcal{D}}_{k}}^{{t}_{c}}left(cdot right)). (xi), (alpha), (tau) are control parameters for the ({mathcal{U}}_{{mathcal{D}}_{j}}^{BES}left(cdot right)). For the({mathcal{D}}_{j}), ({mathcal{J}}_{{mathcal{D}}_{j}}) is the allocated spectrum amount in the ({mathbb{G}}_{{mathcal{W}}_{i}}) game. If ({mathcal{D}}_{j}in {mathbb{Q}}_{{mathcal{W}}_{i}}), the ({mathcal{D}}_{j}) has already obtained the ({Upsilon}_{{mathcal{D}}_{j}}) in the ({mathbb{G}}_{mathfrak{B}}) game. In this case, the ({mathcal{J}}_{{mathcal{D}}_{j}}) is additionally added, and the ({mathcal{D}}_{j}) has total (left({mathcal{J}}_{{mathcal{D}}_{j}}+{Upsilon}_{{mathcal{D}}_{j}}right)) spectrum amount for its QPSs. If ({mathcal{D}}_{j}in {mathbb{B}}_{{mathcal{W}}_{i}}), the ({mathcal{D}}_{j}) gets the ({Upsilon}_{{mathcal{D}}_{j}}) for its BESs. From the ({mathbb{G}}_{{mathcal{W}}_{i}}), we can think the different type services, i.e., BESs and QPSs, are bargaining units. In this case, the ZGBS is preferred to reach a consensus in the ({mathbb{G}}_{{mathcal{W}}_{i}}) game. It is given by:

$$begin{aligned} & ZGBSleft({mathbb{B}}_{{mathcal{W}}_{i}},{mathbb{Q}}_{mathfrak{B}},{mathcal{U}}_{{mathcal{D}}_{j}in left{{mathbb{B}}_{{mathcal{W}}_{i}}cup {mathbb{Q}}_{{mathcal{W}}_{i}}right}}(cdot)right)=underset{left[ldots ,{mathcal{J}}_{{mathcal{D}}_{j}},ldots right]}{user2{max}}prod_{c=1}^{2}left(prod_{{mathcal{D}}_{j}in {Gamma }_{c}}{mathcal{U}}_{{mathcal{D}}_{j}}(cdot)right)\ &quad {text{s.t.}},{ Gamma }_{1}={mathbb{Q}}_{{mathcal{W}}_{i}}, { Gamma }_{2}={mathbb{B}}_{{mathcal{W}}_{i}} {text{ and }}{mathcal{U}}_{{mathcal{D}}_{j}}left(cdot right)=left{begin{array}{c}{mathcal{U}}_{{mathcal{D}}_{j}}^{QPS}left(cdot right),quad {text{if }}{mathcal{D}}_{j}in {Gamma }_{1} \ \ {mathcal{U}}_{{mathcal{D}}_{j}}^{BES}left(cdot right),quad {text{if}} {mathcal{D}}_{j}in {Gamma }_{2} end{array}right.\ end{aligned}$$


Main steps of our LTE/WiFi spectrum allocation algorithm

In this study, we propose a new spectrum allocation algorithm to characterize the LTE/WiFi system platform. To reach the best fair-efficient solution, we adopt the concepts of cooperative bargaining games, and implement the NBS, PNBS and ZGBS. For the negotiation between the (mathfrak{B}) and ({mathcal{W}}_{i}), the ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}), ({mathbb{G}}_{mathfrak{B}}) and ({mathbb{G}}_{{mathcal{W}}_{i}}) games are developed for BESs and QPSs. During discrete time periods, these games are operated repeatedly in a step-by-step online manner. Owing to the desirable characteristics of NBS, PNBS and ZGBS, the ({mathcal{S}}_{mathfrak{B}}) and ({mathcal{S}}_{{mathcal{W}}_{i}}) can be effectively shared under dynamic LTE/WiFi network environments. In the proposed scheme, we do not focus on trying to get an optimal solution based on the traditional optimal approach. Instead, the decision mechanism in our interactive bargaining model is implemented with a polynomial complexity. The main steps of our proposed algorithm can be described as follows, and they are described by Fig. 2:

Fig. 2
figure 2

Flowchart of proposed algorithm

  • Step 1: The values of control parameters are listed in Table 2, and the simulation testbed is given in Sect. 4.

  • Step 2: At each time epoch, multiple devices in the coverage region of the LAA-LTE macro-cell generate their communication tasks. To support these services, the (mathfrak{B}) and ({mathcal{W}}_{1le ile m}) negotiate with each other in the step-by-step interactive and parallel fashion.

  • Step 3: At the first phase, the ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}) is designed to partition the ({mathcal{S}}_{mathfrak{B}}) into ({mathcal{R}}_{mathfrak{B}}) and ({mathcal{R}}_{{mathcal{W}}_{1le ile m}}), which are allocated (mathfrak{B}) and ({mathcal{W}}_{i}), respectively. For this partitioning problem, the NBS is adopted.

  • Step 4: In the ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}) game, utility functions of (mathfrak{B}) and ({mathcal{W}}_{i}) are defined according to (6), and ({mathcal{R}}_{mathfrak{B}}) and ({mathcal{R}}_{{mathcal{W}}_{i}}) are decided using (1) and (7).

  • Step 5: At the second phase, the ({mathbb{G}}_{mathfrak{B}}) game is implemented to share the ({mathcal{R}}_{mathfrak{B}}) among devices in the ({mathbb{Q}}_{mathfrak{B}}). For this game, the PNBS is preferred for the solution of ({mathbb{G}}_{mathfrak{B}}).

  • Step 6: In the ({mathbb{G}}_{mathfrak{B}}) game, the ({mathcal{D}}_{k}in {mathbb{Q}}_{mathfrak{B}}) is a game player, and its utility function is defined according to (8). By using (2) and (9), the ({Upsilon}_{{mathcal{D}}_{k}}) value is obtained.

  • Step 7: At the third phase, each ({mathcal{W}}_{i}) distributes its ({mathcal{R}}_{{mathcal{W}}_{i}}) and ({mathcal{S}}_{{mathcal{W}}_{i}}) for its corresponding devices through the ({mathbb{G}}_{{mathcal{W}}_{i}}) game. It is operated in a parallel fashion for each individual ({mathcal{W}}_{1le ile m}).

  • Step 8: In the ({mathbb{G}}_{{mathcal{W}}_{i}}) game, the ({mathcal{D}}_{j}in left{{mathbb{B}}_{{mathcal{W}}_{i}}cup {mathbb{Q}}_{{mathcal{W}}_{i}}right}) is a game player, and its utility function is defined by using (10). Based on the idea of ZGBS, the ({mathcal{J}}_{{mathcal{D}}_{j}}) value is estimated according to (5) and (11).

  • Step 9: The ({mathbb{G}}_{mathcal{W}}^{mathfrak{B}}), ({mathbb{G}}_{mathfrak{B}}) and ({mathbb{G}}_{{mathcal{W}}_{i}}) games work together to achieve mutual advantages. Constantly, the (mathfrak{B}) and ({mathcal{W}}_{i}) self-monitor the current LTE/WiFi system environments. Proceed to Step 2 for the next game process.

Table 2 System parameters used in the simulation experiments

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