The fundamental functional equation along with some basic preliminary results regarding its investigation are presented in Section 4. In Section 5 we provide expressions for the pgf of the joint stationary queue length distribution at user queues in terms of a solution of a Riemann boundary value problem, while in Section 6 we provide exact bounds for the expected number of backlogged packets at each user queue, without using the theory of boundary value prob- lems.
Numerical results are given in Section 7 and show insights in system performance. Some conclusions and future directions are presented in Section 8. The time is slotted and packets have equal length. The transmission time of a packet corresponds to a single slot. We consider a collision based channel model, and thus, if both users attempt to transmit simultaneously, a collision occurs and both transmissions fail. In such case, the packets have to be re-transmitted in a latter slot.
In particular, at the beginning of a slot, signals are generated in user Uk with probability sk. A signal arriving in an empty buffer has no effect. In this work, we assume that if a signal and a packet transmission occur simultaneously, the signal occurs first. In case of a signal generation, a temporarily network malfunction occurs, and both users remain silent during the slot. Moreover, packet arrivals are scheduled at the end of the slot, early departure late arrival model.
Let Qk,n be the queue length at the buffer of Uk at the beginning of time slot n. Clearly, under usual assumptions Q is irreducible and aperiodic. When a signal is not generated, the packet it will be transmitted successfully to the destination if the node will attempt to transmit and collision will not happen. If a signal is generated, there are two options, either the packet will be dropped from the system or it will be transferred to the other queue.
Thus, it is important to emphasize that the stability region is different than the stable throughput region. In fact, the stable throughput region is a subset of the stability region as we will see in this section.
Thus, the queues are coupled. We will bypass this difficulty by applying the stochastic dominance technique to obtain the exact stability region and the stable throughput region. We use the following definition of queue stability [79, 5]: Definition 1.
Denote by Qti the length of queue i at the beginning of time slot t. If the average arrival rate is greater than the average service rate, then the queue is unstable and the value of Qti approaches infinity almost surely. Theorem 3. To determine the stability region of our system we apply the stochastic dominance technique [64], i. Under this approach, we consider the R1 , and R2 -dominant systems.
Thus, in R1 , node 1 never empties, and hence, node 2 sees a constant probability that a packet will be removed from its queue, while that probability for node 1 depends on the state of node 2, i. We proceed with dominant system R1. The probability m1 of the first node is given by 3. Then, we have the stability region R1 given by 3. For a detailed treatment of dominant systems please refer to [64].
An important observation made in [64] is that the stability conditions obtained by the stochastic dominance technique are not only sufficient but also necessary for the stability of the original system. The indistinguishability argument [64] applies here as well.
Based on the construction of the dominant system, we can see that the queue sizes in the dominant system are always greater than those in the original system, provided they are both initialized to the same value and the arrivals are identical in both systems.
Therefore, the original and the dominant system are indistinguishable at the boundary points. Following the same methodology of dominant systems in the previous theorem, we con- struct two hypothetical systems, R1 and R2.
In the proof of Theorem 3. After replacing 3. Similarly we obtain T2 which is given by 3. Remark 1. Remark 2. In Appendix A we provide an alternative way to derive the stability condition based on general results regarding two-dimensional random walks [77]. In order to proceed with the investigation of the functional equation we also need some crucial preliminary results that are also given below.
Note that some interesting relations can be directly derived by the functional equation 4. Note that 4.
Just for the sake of clarity, note that the left hand side in the first of 4. Note that such an arrival may occur only in case U2 is not empty, i. According to Theorem 2. To show this in the general case, we need some extra assumptions see [3], Sections II. With that in mind, we will focus on the symmetrical case see next Section. For such a case Theorem 2.
S1 , S2 are both simple and smooth. Therefore, the solution of 4. This problem will be transformed in the next section into a Riemann for all b boundary value problem. Moreover, for the stability conditions see Theorem 3.
We proceed as in [3], sections II. In particular, for the contours S1 , S2 , and due to the symmetry of our system the following hold: 1. Then, theorem 3. In view of 5. In particular, setting in 4. We focus only on user U1. Similar expressions can be derived for user U2. Indeed, the properties of the conformal mappings imply that the inverse of these mappings do exist.
Then, the first in 5. Note also that the expressions in 5. Clearly, 5. Several techniques, such as trapezoidal rule, and standard iteration procedures has shown rapid convergence based on the values of the parameters. For a detailed treatment of how we treat numerically 5. Therefore, although the theory of boundary value problems provides a robust mathematical background to obtain expressions for the pgf of the stationary joint queue length distribution at user buffers, it is not an easy task to obtain numerical results; see [3], part IV.
In the next section we provide an efficient approach to derive basic performance metrics without calling for the advanced concepts of theory of boundary value problems. Note that such a case would result in further numerical difficulties. Setting in 4. The following theorem summarizes the main result of this section.
Theorem 6. Remark 3. We consider the case where the users have the same link characteristics and transmission probabilities to facilitate exposition clarity. As we mentioned in that section, the stable throughput region is a subset of the stability region. As si increases, the stability region is becoming slightly broader, this is expected since less packets are transmitted thus we have fewer collisions.
However, for the stable throughput region, we observe that for the low arrival rate for the one user and the high arrival rate for the other user we got less achievable stable throughout, however, in the medium ones we can achieve a larger region. Thus, by enhancing load balancing we can achieve tighter bounds. Moreover, it is easily realized that the increase in transmission probability will decrease the expected number of buffering packets.
This means that the increase in packet arrivals can be balanced by the increase in signal generation, and especially when the probability of a deleting signal i.
The blue lines are for the stability region and the red ones for the stable throughput region. For this reason we avoid considering the case of negative signals, and assuming only triggering signals, i. Our aim is to see when load balancing improves the system performance. In particular, by assuming the the RAG network with only triggering signals i.
Clearly, the network performance can be improved further when assuming deleting signals. Thus, the effect of triggering signals will result in better performance and will improve the quality of service in a multiple access network. For this interacting network of queues, we obtained both the stability SR and the stable throughput STR regions using the stochastic dominance approach.
Moreover, we provided a compact mathematical analysis and obtain expressions for the pgf of the stationary joint queue length distribution at user queues with the aid of the theory of Riemann boundary value problems. Numerical results were also obtained and shown useful insights. Future extensions of this work include the consideration of more realistic models for the wireless channels such as the erasure and the multi packet reception channel.
In addition, security in Internet-of-Things networks can be studied with the consideration of G-networks. Network-level cooperative networks is another interesting direction to be considered. Appendices A An alternative way to derive stability condition In the following we derive the stability condition based on Theorem 6. We d d start with some necessary notation.
We will state only the most interesting part of the ergodicity theorem. For a complete discussion see [77]. Then, following Theorem 6. References [1] G. Fayolle and R. Fayolle, R. Iasnogorodski, and V. Malyshev, Random walks in the quarter-plane: Al- gebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics. Springer-Verlag, Berlin, Cohen and O.
Boxma, Boundary value problems in queueing systems analysis. Seminar on Computer Networking and Performance Evaluation, pp. Pappas, M. Kountouris, J. Jeon, A. Ephremides, and A. Dimitriou and N. Cham: Springer International Publishing, Guillemin and D. Borst, M. Gelenbe and R. Fourneau and E. Gelenbe and A. Fourneau and Y. Henderson, B. Northcote, and P. Chao and M. Harrison and E. Boucherie and O. Jain and K. Artalejo and A. Stochastic Models, vol.
G-Networks and their Applications. Gelenbe and C. E, Statistical, nonlinear, and soft matter physics, vol. Kim and E. Biology Bioinform. Then, the first in 5. Note also that the expressions in 5. Clearly, 5. Several techniques, such as trapezoidal rule, and standard iteration procedures has shown rapid convergence based on the values of the parameters.
For a detailed treatment of how we treat numerically 5. Therefore, although the theory of boundary value problems provides a robust mathematical background to obtain expressions for the pgf of the stationary joint queue length distribution at user buffers, it is not an easy task to obtain numerical results; see [3], part IV. In the next section we provide an efficient approach to derive basic performance metrics without calling for the advanced concepts of theory of boundary value problems.
Note that such a case would result in further numerical difficulties. Setting in 4. The following theorem summarizes the main result of this section. Theorem 6. Remark 3. We consider the case where the users have the same link characteristics and transmission probabilities to facilitate exposition clarity.
As we mentioned in that section, the stable throughput region is a subset of the stability region. As si increases, the stability region is becoming slightly broader, this is expected since less packets are transmitted thus we have fewer collisions. However, for the stable throughput region, we observe that for the low arrival rate for the one user and the high arrival rate for the other user we got less achievable stable throughout, however, in the medium ones we can achieve a larger region.
Thus, by enhancing load balancing we can achieve tighter bounds. Moreover, it is easily realized that the increase in transmission probability will decrease the expected number of buffering packets. This means that the increase in packet arrivals can be balanced by the increase in signal generation, and especially when the probability of a deleting signal i. The blue lines are for the stability region and the red ones for the stable throughput region.
For this reason we avoid considering the case of negative signals, and assuming only triggering signals, i. Our aim is to see when load balancing improves the system performance. In particular, by assuming the the RAG network with only triggering signals i.
Clearly, the network performance can be improved further when assuming deleting signals. Thus, the effect of triggering signals will result in better performance and will improve the quality of service in a multiple access network. For this interacting network of queues, we obtained both the stability SR and the stable throughput STR regions using the stochastic dominance approach. Moreover, we provided a compact mathematical analysis and obtain expressions for the pgf of the stationary joint queue length distribution at user queues with the aid of the theory of Riemann boundary value problems.
Numerical results were also obtained and shown useful insights. Future extensions of this work include the consideration of more realistic models for the wireless channels such as the erasure and the multi packet reception channel. In addition, security in Internet-of-Things networks can be studied with the consideration of G-networks.
Network-level cooperative networks is another interesting direction to be considered. Appendices A An alternative way to derive stability condition In the following we derive the stability condition based on Theorem 6. We d d start with some necessary notation. We will state only the most interesting part of the ergodicity theorem. For a complete discussion see [77]. Then, following Theorem 6. References [1] G. Fayolle and R. Fayolle, R. Iasnogorodski, and V. Malyshev, Random walks in the quarter-plane: Al- gebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics.
Springer-Verlag, Berlin, Cohen and O. Boxma, Boundary value problems in queueing systems analysis. Seminar on Computer Networking and Performance Evaluation, pp. Pappas, M. Kountouris, J. Jeon, A. Ephremides, and A. Dimitriou and N. Cham: Springer International Publishing, Guillemin and D. Borst, M. Gelenbe and R. Fourneau and E. Gelenbe and A. Fourneau and Y. Henderson, B. Northcote, and P. Chao and M. Harrison and E. Boucherie and O. Jain and K.
Artalejo and A. Stochastic Models, vol. G-Networks and their Applications. Gelenbe and C. E, Statistical, nonlinear, and soft matter physics, vol. Kim and E. Biology Bioinform.
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Ephremides and B. Tong, V. Naware, and P. Rao and A. Tsybakov and V. Luo and A. Naware, G. Mergen, and L. Behroozi-Toosi and R. Georgiadis, L. Merakos, and P. Chen, N. Kountouris, and V. Pappas, and M. Pappas, I. Dimitriou, and Z. Kountouris, A. Fayolle, V. Malyshev, M. Menshikov, et al.
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