On the Role of Finite Queues in Cooperative Cognitive Radio Networks with Energy Harvesting

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1 On the Role of Finite Queues in Cooerative Cognitive Radio Networks with Energy Harvesting Mohamed A. Abd-Elmagid, Tamer Elatt, and Karim G. Seddik Wireless Intelligent Networks Center (WINC), Nile University, Giza, Egyt. Det. of EECE, Faculty of Engineering, Cairo University, Giza, Egyt. Electronics and Communications Engineering Deartment, American University in Cairo, AUC Avenue, New Cairo 11835, Egyt. Abstract This aer studies the roblem of cooerative communications in cognitive radio networks where the secondary user is equied with finite length relaying queue as well as finite length battery queue. The major hurdle towards fully characterizing the stable throughut region stems from the sheer comlexity associated with solving the two-dimensional Markov Chain (MC) model for both finite queues. Motivated by this, we relax the roblem and focus on two energy constrained systems, namely, finite battery queue with infinite relay queue and finite relay queue with infinite battery queue. We characterize the stable throughut regions for the two roosed simler systems. For each roosed system, we investigate the imum service rate of the cognitive node subject to stability conditions. Desite the comlexity of the formulated otimization roblems attributed to their non-convexity, we exloit the roblems structure to transform them into linear rograms. Thus, we manage to solve them efficiently using standard known linear rogramming solvers. Our numerical results reveal interesting insights about the role of finite data queues as well as energy limitations on the network erformance, comared to baselines with unlimited energy sources and infinite data queues. I. INTRODUCTION One of the rominent challenges in wireless communication networks is to efficiently utilize the sectrum. The cognitive radio technology is one aroach to tackle the hurdle of sectrum scarcity. In cognitive radio networks, the unlicensed users (secondary users (SUs)) are allowed to exloit the unused sectrum by the licensed users (the rimary users (PUs)) to imrove the utilization of the sectrum [1], [2]. Nevertheless, the sectrum occuation by the SUs is tied with a minimum quality of service guaranteed for the PUs. Cooerative cognitive radio networks has recently attracted considerable attention [3] [5]. [3] introduced a full cooeration rotocol in a wireless multile-access system for a system comosed of N users in which each user is a source and at the same time a otential relay. roosed a cooerative strategy with robabilistic relaying. In this strategy, the SU is equied with two infinite length queues; one is for storing its own ackets and the other is for relaying the PU ackets. If the PU s acket is not successfully decoded by the destination, whereas it is successfully decoded by the SU, the SU admits the PU s acket with robability a. On the other hand, when the PU is sensed idle, the SU serves its own data queue with robability This work was suorted in art by the Egytian National Telecommunications Regulatory Authority (NTRA). b or the relaying queue with robability 1 b. Furthermore, [5] characterized the throughut region when the relaying buffer at the secondary user has finite length. In, [5], it was imlicitly assumed that the SU is equied with unlimited energy suly, i.e., the SU can access the channel whenever the PU is inactive without any energy limitations. Differently from, [5], in this work, we study the scenario when the SU is equied with limited energy source. Furthermore, we investigate the effects caused by the finiteness of queue lengths for both the relaying queue as well as the battery queue. It can be contemlated that the roosed system model constitutes an imortant ste towards real systems. Our main contribution in this aer is three-fold. First, we show the challenges of fully characterizing the stable throughut region when having finite relaying and battery queues. Second, we characterize the stable throughut region for two energy constrained systems, namely, finite battery queue with infinite relay queue and finite relay queue with infinite battery queue. Third, we formulate two otimization roblems to investigate the imum achievable throughut of the SU, subject to queue stability conditions, for the two simler systems. Desite the comlexity of the formulated otimization roblems caused by their non-convexity, we exloit the roblems structure to cast them as linear rograms. This, in turn, leads to efficiently solve the formulated otimization roblems using standard otimization tools. Our numerical results reveal interesting insights about the effects of finite relay and energy queues as well as the energy limitations on the achievable stable throughut region. II. SYSTEM MODEL In this aer, we study cooerative cognitive radio network as shown in Fig. 1. The network consists of a PU and a SU transmitting their ackets to a common destination d. The PU is equied with an infinite queue (Q ) for storing its data ackets. On the other hand, the SU is equied with an infinite queue (Q s ) for storing its data ackets and a finite queue (Q s ) of length L s = N for storing ackets overheard from the PU. The arrival rocesses at the data queues, Q and Q s, are modeled as ernoulli rocesses with means λ and λ s [6], resectively, where λ,λ s 1. The arrival rocesses at both users are assumed to be indeendent of each other, and are indeendent and identically distributed across

2 s Q, L M Q s SU, 1, 2, Q, L N s s,1 1,1 2,1 f s Q PU f d d,2 1,2 2,2 Fig. 1. System model. Fig. 2. Discrete time two-dimensional MC model for Q s and Q, where M = N =2. time slots. It is assumed that the SU is equied with radio frequency energy harvesting circuitry. The harvested energy from the surrounding environment is stored in a finite battery queue (Q ) of length L = M. In addition, the harvested energy is assumed to be harvested in a certain unit and one energy unit is consumed for one transmission attemt. The energy harvesting rocess at the SU is modeled as a ernoulli rocess with mean δ, where δ 1. Time is slotted and one slot duration is equal to one acket transmission time. It is assumed that the SU has erfect sensing. Therefore, the system is collision-free since at most one user transmits one acket each time slot. For a successful transmission, the entire transmitted acket must be received at the destination without error. In addition, the channel must not be in outage, i.e., the received signal-to-noise ratio (SNR) at the destination must not be less than a re-secified threshold required to successfully decode the received acket. Let f d, and f s denote the robability of successful transmission between the PU and destination, the SU and destination, and the PU and SU, resectively. We assume that f d < throughut the aer. This assumtion characterizes the effective relaying role of the SU for the PU overheard ackets in cooerative cognitive radio networks. Moreover, it is assumed that acknowledgement ackets (ACKs) are sent either by the destination for successfully-decoded ackets from the PU or SU, or by the SU for successfully-decoded overheard ackets from the PU. These ACKs are assumed to be instantaneous, error-free and can be heard by all the nodes in the network. The roosed channel access olicy is as follows. It is assumed that the PU has the riority to transmit a acket whenever Q is non emty. If the acket is successfully decoded by the destination, the destination sends back an ACK heard by both users (PU and SU). Therefore, the acket is droed from Q and exits the system. If the acket is not successfully decoded by the destination but successfully decoded by the SU, Q s either admits the acket with robability a i,j or discards it with robability (1 a i,j ), i =,, M and j =,, N. The acket admission robabilities deend on the number of ackets in Q and Q s, i.e., a i,j is the admission robability when Q has i ackets and Q s has j ackets. This admission strategy, in turn, constitutes the robabilistic admission relaying olicy. If the acket is buffered in Q s, the SU sends back an ACK to announce successful recetion of PU s acket. Thus, the acket is droed from Q and the SU becomes resonsible of delivering the PU s acket to the destination. Finally, if the acket is neither successfully decoded by the destination nor decoded by the SU or decoded but not admitted to Q s, then the acket is ket at Q for retransmission in the next time slot. When the PU is idle, the SU s acket transmission deends on the battery and data queues status. If the battery queue is emty, then the SU is unable to transmit a acket. On the other hand, if the battery queue is not emty, the SU either transmits a acket from Q s with robability b i,j or from Q s with robability (1 b i,j ), i =,,M and j =,,N. Also, we notice that the queue selection robability deends on the number of ackets in Q and Q s. If the destination successfully decodes the acket, it sends back an ACk heard by the SU. Therefore, the acket is droed from either Q s or Q s and exits the system. Otherwise, the acket is ket at its queue for later retransmission. In the next section, we characterize the stability conditions of all infinite queues in the network. III. STALE THROUGHPUT REGION In this section, we characterize the stable throughut region of the roosed system model. The system is stable if all of its queues are stable. Loyens theorem [7] rovides the stability condition for an infinite size queue. The theorem states that if the queue arrival and service rocesses are stationary, the queue is stable if and only if the acket arrival rate is strictly less than the acket service rate. Note that Q and Q s are finite queues; therefore, the number of ackets in each of them will never grow to infinity since it is uer bounded by M and N, resectively. A acket leavesq if it is either successfully decoded by the destination or successfully decoded by the SU and admitted to the relaying buffer (Q s ). Therefore, the service rate of Q is given by 2

3 X M = f d +(1 f d )f s a i,j π i,j, (1) where π i,j denotes the steady state robability that Q has i ackets and Q s has j ackets at a given time slot, i =,1,,M and j =,1,,N. Therefore, the stability condition for Q is given by λ <f d +(1 f d )f s M X a i,j π i,j. (2) Similarly, a acket leaves Q s if Q is emty which has a robability of 1 λ, Q is not emty or emty but there is an energy acket arrival, Q s is selected for transmission which haens with robability b i,j and the destination successfully decodes the acket which has a robability of. Thus, the stability condition for Q s can be exressed as λ s < 1 b i,j π i,j +δ b,j π,j A. It is worth nothing that the service rate of ackets in both queues Q s and Q s deends on the state of the battery queue (Q ) at the secondary user and vise versa. This deendency, in turn, leads to an interacting system of queues and comlicates the characterization of the stable throughut region. In sequel, we use the Dominant System concet [8] in order to tackle this roblem such that we assume that Q s and Q s have dummy ackets to transmit when they are emty and, hence, the service rate of Q becomes only deendent on the PU s status. This system stochastically dominates our system since the SU queues lengths in the dominant system are always larger than that of our system if both systems start from the same initial state, have the same arrivals and encounter the same acket losses. Now, our objective is to calculate the steady state distribution of Q and Q s (π i,j ) in order to fully characterize the stable throughut region. And, hence, be ready to investigate the otimal admission and selection robabilities (a i,j andb i,j ) in order to achieve the imum SU s service rate subject to queue stability conditions. We start by showing the comlexity of fully characterizing the steady state distribution, which arises from the nature of discrete time multidimensional MC with diagonal transitions. Afterwards, we relax the assumtion of having both Q and Q s with finite lengths, and study two different settings, namely, finite battery queue with infinite relay queue and finite relay queue with infinite battery queue, with comletely characterizing of their stable throughut regions. Q and Q s can be modeled as a discrete time twodimensional MC. The MC is shown in Fig. 2 where state i,j denotes that the number of ackets in Q and Q s are i and j, (3) resectively. The robability of moving from state i,j to state i+1,j+1 is the robability that Q is non emty, an energy acket arrives at Q, the PU s acket is not successfully decoded at the destination, the SU successfully decodes the acket and Q s admits the acket. Hence, P i,j i+1,j+1 can be exressed as P i,j i+1,j+1 = λ δ(1 f d )f s a i,j. (4) Similarly, the robability of moving from state i,j to state i 1,j 1 is the robability thatq is emty, there is no energy acket arrival at Q, Q s is selected for transmission and the acket is successfully decoded at the destination. Therefore, P i,j i 1,j 1 is given by P i,j i 1,j 1 = (1 δ) (1 b i,j ). (5) Using the same rationale, the remaining transition robabilities can be exressed as follows P i,j i+1,j = λ δ(f d +(1 f d )(1 f s a i,j )), (6) P i,j i 1,j = P i,j i,j+1 = λ (1 δ)(1 f d )f s a i,j, (7) P i,j i,j 1 = δ (1 b i,j ), (8) (1 δ)(b i,j +(1 b i,j )(1 )). (9) Indeed, the existence of diagonal transitions, i.e., P i,j i 1,j 1 and P i,j i+1,j+1, comlicates the solution of the balance equations along with the normalization equation P i=m P j=n π i,j =1. Consequently, the roductfrom solution of the MC is inalicable and there is no closed form exressions for the steady state robabilities. It is worth mentioning that various aroximation techniques for solution of multidimensional MCs have been extensively used in the literature [9], e.g., equilibrium oint analysis (EPA). However, these aroaches deend on aroximating the stationary robability distribution of the MC by a unit imulse located at a oint in the state sace where the system is in equilibrium. Thus, they do not fit with our objective of having closed form exressions for the steady state robabilities to be able to otimally tune the admission and selection robabilities to achieve the imum SU s service rate subject to stability conditions. Motivated by this, we relax the assumtion of both Q and Q s having finite lengths. More secifically, we focus on two different energy constrained systems, namely, finite battery queue with infinite relay queue and finite relay queue with infinite battery queue, in the following two subsections. 3

4 A. Finite battery queue with infinite relay queue Under this setting, we assume that Q remains with finite length M, but Q s becomes an infinite queue. Note that the admission and selection robabilities (a i and b i for i =,,M) become only deendent on the state of Q. It is worth nothing that the stability conditions for Q and Q s, given by (2) and (3), will reduce to the following exressions λ s < X M λ <f d +(1 f d )f s a i πi, (1) M X b i π i +δb π, (11) resectively, where π i is the steady state robability that Q has i energy ackets at a given time slot. y alying Loyens theorem, the stability condition for Q s can be derived as follows. A acket is buffered at Q s if Q is not emty which haens with robability λ. In addition, the acket is not successfully decoded by the destination which haens with robability 1 f d, whereas it is successfully decoded by the SU which haens with robability f s and is admitted to Q s which has a robability of a i, i =,1,,M. Thus, λ s is given by λ s = λ (1 f d )f s MX a i πi. (12) On the other hand, a acket leaves Q s if Q is emty which haens with robability 1 λ, Q is not emty or emty but there is an energy acket arrival, Q s is selected for transmission which haens with robability 1 b i, i =,1,,M, and the acket is successfully decoded at the destination which has a robability of. Therefore, and the stability condition for Q s are given resectively by = X M (1 b i )πi +δ(1 b )π, (13) λ s <. (14) With the urose of fully characterizing the stability region, we shall now calculate the steady state robabilities ofq (π i for i =,,M). Q can be modeled as a discrete time M M 1 M. The MC is shown in Fig. 3 where state i denotes that the number of ackets in Q is i. Let λ i and µ i denote the robability of moving from state i to state i+1 and the robability of moving from state i to state i 1, resectively. λ i is the robability thatq is not emty and an energy acket arrives at Q. On the other hand, µ i is the robability that Q is emty and there is no an arrival energy acket. Thus, using the balance equations, the steady state robabilities of Q are given by 1 M 2 1 M-1 M 1 2 M1 M 1 M Fig. 3. Discrete time MC model for Q in finite battery queue with infinite relay queue system. π i+1 = δ λ µ πi, (15) (1 δ) where i =, 1,, M 1. Alying the normalization condition MX πi =1, (16) along with (15), the steady state distribution of Q can be comletely characterized.. Finite relay queue with infinite battery queue Under this setting, we assume that Q s remains with finite length N, but Q becomes an infinite queue. Although the relay finiteness effects are studied in [5], it was imlicitly assumed that the system has no energy limitation, i.e., the SU always has energy ackets to transmit whenever it has the oortunity to access the channel. On the contrary, in this subsection, we focus on the more interesting ractical scenario of having a limited-energy system. The energy limitation is characterized through the constraint that the energy arrival rate at Q is strictly less than its service rate. Thus, the number of energy ackets inside Q will never grow to infinity and there is always a non-zero robability of having an emty Q. In this scenario, the admission and selection robabilities (a j and b j for j =,,N) become only deendent on the state of (Q s ). We set a N =to revent Q s from admitting any overheard PU s acket when it is full, and b =1 to revent allocating any transmission time slots for Q s when it is emty. The stability conditions for Q and Q s, given by (2) and (3), will reduce to the following exressions λ <f d +(1 f d )f s N X λ s < δ 1 λ a j π s j, (17) b j π s j, (18) 4

5 s s 1 s N 2 1 N-1 N s 1 s 2 s N 1 s N s N 1 Fig. 4. Discrete time MC model for Q s in finite relay queue with infinite battery queue system. resectively, where π s j is the steady state robability that Q s hasj ackets at a given time slot. Note that the fraction δ/ in (18) reresents the robability that Q is caable of suorting the transmission of the SU s acket, and it is the sum of the two robabilities: the robability of having non emty Q and the robability of having an emty Q but there is an energy acket arrival. Finally, the energy limitation constraint is given by δ λ < 1 λ (1 δ). (19) Next, we shall now calculate the steady state distribution of Q s. Q s can be modeled as a discrete time M M 1 N. The MC is shown in Fig. 4 where state j denotes that the number of ackets in Q s is j. Let λ s j and j denote the robability of moving from state j to state j +1 and the robability of moving from state j to state j 1, resectively. λ s j is the robability that Q is not emty, the acket is not successfully decoded by the destination, whereas it is successfully decoded by the SU and is admitted to Q s. On the other hand, j is the robability that Q is emty, Q is caable of suorting the transmission of the SU s acket as discussed in (18), Q s is selected for transmission and the acket is successfully decoded at the destination. Thus, using the balance equations, the steady state robabilities of Q s are given by π s j+1 = λ f s (1 f d )a j π s j δ(1 b j+1 ), (2) where j =, 1,, N 1. Alying the normalization condition π s j =1, (21) along with (2), the steady state distribution of Q s can be comletely characterized. In the next section, we formulate the stable throughut region otimization roblems for the two systems studied in this section and discuss the solution for them. IV. SU S THROUGHPUTMAXIMIZATIONPROLEM In this section, our objective is to investigate the otimal admission and selection robabilities to achieve the imum SU s service rate for both considered energy constrained systems, namely, finite battery queue with infinite relay queue and finite relay queue with infinite battery queue. The imum SU s service rates are investigated subject to the stable throughut regions characterized in Section III. A. Finite battery queue with infinite relay queue The SU s service rate imization roblem for finite battery queue with infinite relay queue system can be formulated as P1 : a i,b i,π i,µ s.t. λ <, M X X M = f d +(1 f d )f s a i πi, λ s <, π i,a i,b i 1,,,M, b i π i +δb π (15),(16), (22) where λ s and are given by (12) and (13), resectively. It is worth nothing that P1 is a non-convex otimization roblem. However, we exloit the roblem structure in order to transform it into a linear rogram. More secifically, by defining the new variables x i = a i π i,y i = b i π i,,,m, (23) P1 reduces into a linear rogram for a given as follows. First, we have the following constraints on the new defined variables x i,y i π i,,,m. (24) Second, we can rewrite the constraint in (14) as MX x i < ( λ ) MX ( ) ( ) π λ f s (1 f d ) i y i +δ π y. (25) Finally, by substituting with the new defined variables into the objective function and the remaining constraints, P1 turns out to be a linear rogram for a given and can be exressed as follows P1 : s.t. x i,y i,π i M X X M = f d +(1 f d )f s x i, π i 1,,,M, (15),(16),(24),(25). y i +δy (26) 5

6 From (1), the feasible values of over which the linear rogram runs are given by (λ,f d ) f d + (1 f d )f s.. Finite relay queue with infinite battery queue The SU s service rate imization roblem for finite relay queue with infinite battery queue system can be formulated as P2 : δ b j π s j a j,b j,π s j,µ s.t. λ <, = f d +(1 f d )f s N X a j π s j, δ λ < (1 δ), a N =,b =1, a j,b j,π s j 1,,,N, (2),(21). (27) y insecting P2, we can easily see that it is a non-convex otimization roblem. However, similar to P1, P2 s structure can be exloited to transform it into a linear rogram. y defining the new variables x j = a j π s j,y j = b j π s j,,,n, (28) P1 reduces into a linear rogram for a given as follows. First, we have the following constraints on the new defined variables x j,y j π s j,,,n. (29) Second, we can rewrite the constraint in (2) as j+1 y j+1 = λ f s (1 f d ) x j,,,n 1. (3) δ π s Finally, by substituting with the new defined variables into the objective function and the remaining constraints, P2 turns out to be a linear rogram for a given and can be exressed as follows P2 : s.t. x j,y j,π s j δ N X y j X N = f d +(1 f d )f s x j, x N =,y = π s, π s j 1,,,N, (21),(29),(3). (31) From (17) and (19), the feasible values of over which the linear rogram runs are given by ( λ 1 δ,f d) f d +(1 f d )f s. (32) V. NUMERICAL RESULTS In this section, we rovide numerical results showing the merits of the formulated otimization roblems and the associated trade-offs. Motivated by the convexity of the roosed linear rograms, we use standard otimization tools, e.g., CVX [1], to obtain the otimal solution. If not otherwise stated, we use the following arameters f d =.3, f s =.4, =.8 and δ =.5. Our objective is to comare the erformance of our roosed cooerative cognitive radio network systems with limited energy sources with that of the baseline cooerative cognitive radio networks with non limited energy sources introduced in, [5]. In Fig. 5, Our objective is to show the effect of the arrival rate of the harvested energy ackets at the SU on the stable throughut region for finite battery queue with infinite relay queue system (P1). Towards this objective, we fix M = 1 and lot the stable throughut region for different values of δ (δ =.3,.5,.7 and.9). It is observed that as the average arrival rate of the harvested energy ackets increases, the throughut region exands. This is attributed to the fact that as the average arrival rate of harvested energy ackets increases, the likelihood that Q is emty decreases. This, in turn, manages the SU to achieve larger service rate ( ) for a given arrival data ackets rate of the PU (λ ). It is further observed that the stable throughut region obtained by P1 aroaches to that of with non limited energy sources as δ increases. Fig. 6 comares the achievable stable throughut region of the baseline cooerative cognitive radio with that of P1 for different values of battery queue length (L =1, 3, 1 and 15). It is observed that the stable throughut region obtained by P1 exands as M increases. Increasing M results in having a lower robability for both Q is emty and SU s harvested energy ackets losses. Therefore, the service rate of the SU increases with M. However, even for large values of Q length (M = 15), the imum achievable service rate by P1 is half that of the baseline sytem with no energy limitations for λ =. This, in turn, highlights the imact of the robabilistic arrival of energy ackets at the SU (δ =.5). In Fig. 7, we comare the stable throughut region achieved by finite relay queue with infinite battery queue system (P2), for different values of δ and N = 1, with the baseline system with infinite length Q s and no energy limitations, and the system studied in [5] with finite length Q s and no energy limitations. It is observed that the ractical energy limitation constraint in P2 greatly influences the achievable throughut region, comared to the scenario of no energy limitations [5]. More secifically, from (32), the energy limitation constraint restricts P2 feasibility to be only feasible for λ < (1 δ)(f d +(1 f d )f s ). This, in turn, leads to a trade-off between achieving high SU s service rate for small values of λ and wide range of λ values over which P2 is feasible (as a function of δ). Fig. 8 comares the achievable stable throughut region of P2 with that of, [5] for different values of Q s length 6

7 P1 (M = 1, δ =.3) P2 (N = 1, δ =.3).8.7 P1 (M = 1, δ =.5) P1 (M = 1, δ =.7) P1 (M = 1, δ =.9).8.7 P2 (N = 1, δ =.5) P2 (N = 1, δ =.7) P2 (N = 1, δ =.9) [5] (N = 1) λ λ Fig. 5. The stable throughut region of finite battery queue with infinite relay queue system for different values of average arrival energy ackets er time slot (δ). Fig. 7. The stable throughut region of finite relay queue with infinite battery queue system for different values of average arrival energy ackets er time slot (δ)..9.9 P2 (N = 1, δ =.5) P1 (M = 1, δ =.5) P1 (M = 3, δ =.5) P1 (M = 1, δ =.5) P1 (M = 15, δ =.5) P2 (N = 3, δ =.5) P2 (N = 1, δ =.5) P2 (N = 15, δ =.5) [5] (N = 1) [5] (N = 3) [5] (N = 1) [5] (N = 15) λ λ Fig. 6. The stable throughut region of finite battery queue with infinite relay queue system for different values of Q length. Fig. 8. The stable throughut region of finite relay queue with infinite battery queue system for different values of Q s length. (L s =1, 3, 1 and 15). Desite the increase of Q s length, we observe that the stable throughut regions achieved by N = 3, N = 1 and N = 15 are identical. This haens due to the forced range of λ by the energy limitation constraint over which P2 is feasible (λ <.29). In other words, the energy limitation constraint revents the stable throughut region s exansion caused by increasing Q s length. VI. CONCLUSION In this aer, we studied cooerative cognitive radio network where the secondary user is equied with finite length relaying queue as well as finite length battery queue. Motivated by the comlexity of fully characterizing the stable throughut region, we relaxed the roosed system model and roosed two energy constrained systems. The stable throughut regions were characterized for our roosed systems. We formulated the stable throughut region otimization roblem for each roosed system and showed how to solve it. Finally, we comared the achievable throughut region by each roosed system with that of the baseline system with unlimited energy sources and infinite queues. REFERENCES [1] J. Mitola, Cognitive radio an integrated agent architecture for software defined radio, PhD dissertation, Royal Institute of Technology (KTH), May 2. [2] S. Haykin, Cognitive radio: brain-emowered wireless communications, IEEE Journal on Selected Areas in Communications, Feb. 25. [3]. Rong and A. Ehremides, Cooerative access in wireless networks: Stable throughut and delay, IEEE Transactions on Information Theory, Set M. Ashour, A. El-Sherif, T. Elatt, and A. Mohamed, Cooerative access in cognitive radio networks: stable throughut and delay tradeoffs, Modeling and Otimization in Mobile, Ad Hoc and Wireless Networks, May 214. [5] A. Elmahdy, A. El-Keyi, T. Elatt, and K. Seddik, On the stable throughut of cooerative cognitive radio networks with finite relaying buffer, Personal Indoor and Mobile Radio Communication, Set [6] J. Jeon and A. Ehremides, The stability region of random multile access under stochastic energy harvesting, IEEE International Symosium on Information Theory Proceedings, July 211. [7] R. M. Loynes, The stability of a queue with non-indeendent interarrival and service times, Mathematical Proceedings of the Cambridge Philosohical Society, [8] R. Rao and A. Ehremides, On the stability of interacting queues in a multile-access system, IEEE Transactions on Information Theory, Se [9] M. Woodward, Equivalence of aroximation techniques for solution of multidimensional markov chains in network modelling, Electronics letters, [1] S. oyd and L. Vandenberghe, Convex otimization. Cambridge university ress, 24. 7