Distributed Power Control and Coding-Modulation Adaptation in Wireless Networks using Annealed Gibbs Sampling
In wireless networks, the transmission rate of a link is determined by received signal strength, interference from simultaneous transmissions, and available coding-modulation schemes. Rate allocation is a key problem in wireless network design, but a very challenging problem because: (i) wireless interference is global, i.e., a transmission interferes all other simultaneous transmissions, and (ii) the rate-power relation is non-convex and non-continuous, where the discontinuity is due to limited number of coding-modulation choices in practical systems. In this paper, we propose a distributed power control and coding-modulation adaptation algorithm using annealed Gibbs sampling, which achieves throughput optimality in an arbitrary network topology. We consider a realistic Signal-to-Interference-and-Noise-Ratio (SINR) based interference model, and assume continuous power space and finite rate options (coding-modulation choices). Our algorithm first decomposes network-wide interference to local interference by properly choosing a “neighborhood” for each transmitter and bounding the interference from non-neighbor nodes. The power update policy is then carefully designed to emulate a Gibbs sampler over a Markov chain with a continuous state space. We further exploit the technique of simulated annealing to speed up the convergence of the algorithm to the optimal power and coding-modulation configuration. Finally, simulation results demonstrate the superior performance of the proposed algorithm.
Wireless communications have become one of the main means of communications over the last two decades, in the form of both cellular (WWAN) and home/business access point (WLAN) communications . Recently, with the development of data centric mobile devices, e.g., iPhone, we have seen a renewed interest in enabling more flexible wireless networks, e.g., ad hoc networks and peer to peer networks . A key problem in the design of ad hoc wireless networks is link-rate control, i.e., controlling transmission rates of the links. In wireless networks, the transmission rate of a link is determined by received signal strength, interference from simultaneous transmissions, and available coding-modulation schemes. Because wireless interference is global, and the rate-power relation is non-convex  and non-continuous, distributed link-rate control in ad hoc wireless networks is a very challenging problem.
One approach to tackle the link-rate control problem in the literature is to assume that the coding-modulation scheme is predetermined, i.e., all links use the same coding-modulation scheme. In this case, there is an SINR threshold associated with each link, and a transmission over the link can be successfully decoded when the actual SINR is above the threshold. This assumption is reasonable for voice-centric wireless networks where the same voice codec is used at all devices. Under this assumption, the link-rate control is translated into a power control problem where the objective is to find a set of minimum transmission powers such that the SINRs at all links are above the thresholds, if possible at all. This problem has been well studied in the context of the power control in cellular communications  and simple iterative algorithms can be shown to converge to the optimal feasible power allocations.
Another approach in the literature is to assume the link rate is a continuous function of the SINR of the link [11, 15]. For example, a model that has been extensively adopted is to assume where is the transmission rate of link In other words, it assumes that for each SINR level, the capacity achieving coding-modulation is available. Under this assumption, the rate control problem again is formulated as a power control problem where the objective is to find a set of powers to maximize system utility defined upon achievable rates where is the utility function associated with link This problem is also well understood in cellular networks given the recent advances in optimal power control and rate assignment[5, 6], where distributed iterative algorithms are shown to converge to the utility maximizing power allocations, after introducing a small signaling overhead to the cellular air interface. However, these approaches ignore the non-convex nature of the problem and the algorithms proposed here converge to the utility maximizing operating point on Pareto boundary of the rate region, assuming all devices have to transmit all the time. In the context of ad hoc networks, such approaches can be highly sub-optimal since the time-sharing, or inter-link scheduling, nature of the problem has to be considered due to highly non-convex nature of the rate-power function. Towards this end, queue-length distributed scheduling is shown to be throughput optimal [7, 9, 14, 12], through the use of MCMC (Markov Chain Monte Carlo) models. These results however assume collision-based interference model, which in general is over-conservative, and assume fixed transmit power and coding-modulation scheme. Both transmit power and coding-modulation can be adaptively chosen in practical systems. For example, in 802.11g, eight rate options are available, and many 802.11 chip solutions have capability of packet to packet power control with very good granularity (0.5dBm). Gibbs sampling based distributed power control algorithms have also been developed in [13, 18]. However, these work again assumes the rate is a continuous function of the SINR level, and ignores the fact that the set of coding-modulation schemes is finite in practical systems.
In this paper, we extend the framework in [7, 9, 14, 12] to develop a distributed joint power control and rate scheduling algorithm for wireless networks based on the SINR-based interference model. We assume that each node has a finite number of coding-modulation choices, but can continuously control transmit power. We propose a distributed algorithm that maximizes the sum of weighted link rates where is the rate of link The main results of this paper are summarized below:
We consider realistic SINR-based interference model, where a transmission interferes with all other simultaneous transmissions in the network. Our algorithm decomposes network-wide interference to local interference by properly choosing a “neighborhood” for each node, and bounding the interference from non-neighbor nodes.
We assume continuous power space and finite coding-modulation choices (rate options). The objective of the algorithm is to find a power and coding-modulation configuration that maximizes the sum of weighted link-rates
where is the queue length of link at time slot 111We use as the link weight so that the algorithm is throughput optimal when problem is solved at each time slot. is a vector containing the power levels of all the links in the network, is the maximum power constraint, and is the coding-modulation scheme. Due to the nonconvexity and discontinuity of optimization problem (1) is very hard to solve in general. Motivated by recent breakthrough of using MCMC to solve MaxWeight scheduling in a completely distributed fashion, we propose a power and coding-modulation update algorithm that emulates a Gibbs sampler over a Markov chain with a continuous state space (the power level of a transmitter is assumed to be continuously adjustable).
The algorithm based on the Gibbs sampling may be trapped in a local-optimal configuration for an extended period of time. To overcome this problem, we exploit the technique of simulated annealing to speed up the convergence to the optimal power and coding-modulation configuration. The convergence of the algorithm under annealed Gibbs sampling is proved. From the best of our knowledge, this is the first algorithm that uses annealed Gibbs sampling in a distributed fashion with continuous sample space and has provable convergence.
Ii System Model
We consider a wireless network with single-hop traffic flows. The network is modeled as a graph where is the set of nodes, and is the set of directed links. Let denote the number of links. We assume that time is slotted. Each transmitter maintains a buffer for each outgoing link if there is a flow over link Note that even if there are multiple flows over link a single queue is sufficient for maintaining the stability of the network. The queue length in time slot is denoted by Each transmitter has limited total transmit power and denotes the transmit power of link at time slot
We assume all links have stationary channels, and each transmitter can tune its transmit power continuously from to but the number of feasible coding-modulation choices is finite. Each coding-modulation associates with a fixed data rate, and a minimum SINR requirement. Thus, the data rate of a link is a step function of the SINR of the link. The SINR of link is
where is the variance of Gaussian background noise experienced by node and is the channel gain from node to node In this paper, all s and s are assumed to be fixed, i.e. we consider stationary channels.
Denote by the transmission rate of at time slot and the number of bits that arrive at the buffer of the transmitter of link at the end of time slot Then, the queue length evolves as following:
Let denote the set of all feasible power configurations of the network, i.e.,
For each link given a power configuration the SINR of the link is determined by equality (2). The transmission rate can be written as where is the coding-modulation scheme.
In this paper, we assume a transmitter always selects the coding-modulation scheme with the highest rate under the given SINR. Each coding-modulation scheme has a minimum requirement on the SINR level. So is a function of and rate can be written as a function of Then, we define as the set of achievable rate vectors under feasible power configurations and modulations, i.e.,
The capacity region of the network is the set of all arrival rate vectors for which there exists a power control algorithm that can stabilize the network, i.e., keep the queue lengths from growing unboundedly. It is well known that the capacity region is :
where is the convex hull of the set of achievable rates with feasible power configurations, and denotes componentwise inequality. A power control and coding-modulation adaptation algorithm is said to be throughput optimal if it can stabilize the network for all arrival rates in the capacity region
It is well-known that if a rate control algorithm can solve the MaxWeight problem  for each time slot, then the algorithm is throughput optimal. The focus of this paper is to develop a power-control and coding-modulation adaptation algorithm to solve the following MaxWeight problem:
Recall that since is a step function of multiple power configurations may result in the maximum weighted sum. We therefore added a penalty function with a small in the objective function so that the algorithm yields a power configuration whose weighted sum-rate is close to the optimal one and its sum power is small. Without this penalty term, the algorithm may result in a solution with maximum sum weighted rate but large This penalty term makes sure the proposed algorithm is energy efficient.
We are interested in obtaining the optimal power, coding, and modulation configuration that maximizes the weighted-sum-rate while minimizing the total transmit power. We can solve this problem by constructing a Markov chain whose state is the power configuration, and the stationary density satisfies
Gibbs sampler is a classical way to construct such a Markov Chain with stationary distribution (6). Given the current state the Gibbs sampler selects a link, say in a predetermined order and changes the transmit power to with probability
where denotes the vector of transmit powers except that of link It can be verified that the stationary distribution of this Markov chain is (6) by detailed balance equation, i.e., Therefore, if the power vector is updated according to this Markov chain, it will converge to with probability one when
There are, however, several difficulties in using Gibbs sampler for distributed rate control in wireless networks.
First, to compute the conditional distribution, link must know the rate of all the links in the network, which incurs significant communication overhead.
Second, the updating sequence of a Gibbs sampler is predefined, which results in the need of a central controller.
Further, when is close to zero,
for such that
In other words, the power configuration may stay in a local optimum for a long period of time. This is in fact a critical weakness of MCMC methods. We will use simulated annealing technique in our algorithm to overcome this weakness.
Iii-a Neighborhood and Virtual Rate
To overcome the global interference, we note that because of channel attenuation, interference caused by a remote transmitter in general is negligible. We therefore define a neighborhood for each node We say a node is a one-hop neighbor of node if i.e., the channel gain is above certain threshold. Denote by the set of one-hop neighbors of node where the superscript indicates it is the set of one-hop neighbors. We further denote by the set of two-hop neighbors of node i.e., node belongs to if and and there exists a node such that and
Then, we define to be
which is called noise+partial-interference at node for link at time slot We further define to be the noise+partial-interference at node if link uses transmit power and to be the maximum noise+partial-interference allowed to achieve the SINR requirement of coding-modulation scheme while link does not change its power level. Let denote where is an upper bound on the interference experienced at node from the non-neighboring transmitters of We assume is known. By including this upper bound in the SINR computation, we guarantee that the SINR of link is a function of its neighbors’ transmit powers and is independent of non-neighbor nodes. This localizes the interference.
Given the definition of noise+partial-interference, the virtual rate of link is defined to be
Observe that although the power level is continuous, and the SINR of neighboring links are continuous, the virtual rate choices are discrete and finite. Suppose is changing its power and link is affected. For each coding-modulation scheme of link let denote the SINR requirement of and denote the corresponding noise+partial-interference requirement. Assuming the transmit powers of all other nodes are fixed, the power level such that is called a critical power, which is highest power node can use for coding-modulation choice to be feasible over link
Iii-B Decision Set
To overcome the issue of predefined update sequence in classic Gibbs sampling, we adopt the technique proposed in  to generate a decision set at the beginning of each time slot.
A decision set is a set of transmitters such that, for any two transmitters and in
Clearly, two transmitters and in the decision set are not one-hop or two-hop neighbors. In the proposed algorithm, only the links in the decision set are allowed to update their transmit powers. By properly generating a decision set, the evolution of power configuration is a reversible Markov chain with stationary density (6).
Iii-C Required Information
We further assume that node has the following knowledge:
the channel gain from node to its one-hop neighbors, i.e., for all
for each link whose receiver is s one-hop neighbor, i.e.,
the virtual transmit power of i.e., where the virtual transmit power is the intermediate power level obtained during each time slot of the proposed algorithm. The real transmit power is updated at the end of every super time slot. So the virtual power is an intermediate result obtained and maintained during the calculation and is not the actual transmit power.
the channel gain of i.e.,
the virtual partial-interference-plus-noise of i.e., 222calculated based on virtual transmit powers
the queue length of i.e.,
the feasible modulations of and the maximum allowed noise+partial-interference, of each modulation
Notice that and change over time. We will explain the way node obtains these values from node in the algorithm. Further we assume all channel gains are known to and are fixed. The feasible modulations and the minimum SINR requirement for each modulation are assumed to be known a-priori, and do not need to be exchanged.
Iii-D Distributed Power Control and Coding Modulation Adaptation Algorithm
We now present the proposed algorithm, where the evolution of power configuration emulates a Gibbs sampler. To improve the convergence of the Gibbs sampler, we exploit the technique of simulated annealing .
We group every time slots into a super time slot. In each super time slot, we run the algorithm for times in the background of each node. In time slot of a super time slot, the value is set to be where is the “temperature” in the terminology of simulated annealing, and is a positive constant that can be tuned to control the convergence of the proposed algorithm. The idea of the simulated annealing is to start with a high temperature (large ) under which the Markov chain mixes rapidly. Then by slowly decreasing the temperature, the state of the Markov chain will converge to the optimal configuration. It has been well-known that annealing can significantly reduce the convergence time. The structures of time slot and super time slot are illustrated in Figure 1 and 2.
All nodes maintain virtual power and the initial power configuration known by all the nodes in the network. The following algorithm describes the process of updating virtual power configuration following an annealed Gibbs sampler. The real transmit power is then determined based on actual SINR.
At time slot of a super time slot, the algorithm works as follows:
Generating decision set: Each time slot consists of control slots at the beginning. A decision set is determined at the end of the control slots. Only the transmitters in the decision set update their virtual power levels at this time slot. At time slot transmitter contends for being in the decision set as follows:
Node uniformly and randomly selects an integer backoff time from and wait for control slots.
If receives an INTENT message from another transmitter such that
before control slot node will not be included in the decision set in this time slot. Here, we assume the INTENT message from has the id of and the signal is strong enough, so that ’s one-hop and two-hop neighbors, e.g., node know this INTENT message is sent by
If node senses a collision of INTENT messages from nodes will not be in the decision set in this time slot.
If node does not receive any INTENT message from its one-hop or two-hop neighbors before control slot , node will broadcast an INTENT message to its one-hop and two-hop neighbors in control slot .
If the INTENT message from node collides with another INTENT message sent by node will not be in the decision set in this time slot.
If there is no collision, node will be included in the decision set in this time slot.
We note that is selected to be large enough so that the collision of the INTENT messages happens with low probability.
Link selection: Let denote the outgoing degree of node i.e., In this step, each transmitter selects an outgoing link to update its virtual power as following:
If there was an active outgoing link such that will update the power of link in time slot with probability
If there was no active outgoing link such that uniformly randomly selects a link from its outgoing links, and then updates its virtual power
Critical power level computation: Node computes the critical power level as follows:
Node computes the critical partial-noise-plus-interference of link corresponding to each
Node computes the critical power of such that when link uses this power level, the resulting partial-noise-plus-interference of link is
(Some modulations of link need very high SINR, which cannot be achieved even link reduces its power to 0. For these modulations, we just let be zero, we will consider the critical power separately in the following step.)
Virtual rates computation: Now for each node in the decision set , it computes the virtual rate of link and the virtual rate of the links whose receiver is s neighbor as following:
Arrange the critical power levels
in ascending order, denoted by
Compute the SINR of link when the power of link is zero:
Further, find the coding-modulation of link with the largest transmission rate corresponding to this SINR. Let the coding-modulations of all the neighboring links of link be denoted by a vector .
Given this initial coding-modulation vector obtains the coding-modulation vector:
corresponding to each critical power
Obtain the rates related to the coding-modulations of neighboring links, when
Note that for each link, the is the coding-modulation scheme with the highest transmission rate assuming node transmits with power
Compute the virtual local weight, under each critical power level
for where is the queue length at the beginning of the super time slot.
Power-level selection: Let be the normalization factor defined as
Node first selects a power interval with following probability:
Suppose the interval is selected, then node randomly selects a virtual power level according to the following probability density function (pdf):
which can be done by using the inverse transform sampling method.
Information exchange: If the virtual power of a node has changed, i.e., then node broadcasts to all its one-hop neighbors. Each neighbor computes the virtual partial-interference-plus-noise of i.e., If node broadcasts to all its one-hop neighbors.
Update real transmit power: At the end of a super time slot, node updates its transmit power for link to be Node then measures the actual SINR and reports to node Node selects the coding-modulation scheme with the highest rate among those such that Packets of flow are transmitted with power and coding-modulation scheme Note that the real transmit powers are updated only once every time slots.
We now present a simple example to show how the algorithm works.
Consider the wireless network depicted in Fig. 3. There are 4 links in the network. Assume the channel gain, background noises and queue lengths are: and The virtual power level of the links in the previous time slot are Further, we assume that there are two feasible coding-modulation schemes for each link: BPSK with rate 1, and QPSK with rate 2, and the SINR requirement for the modulations are 4 and 8, respectively.
In this example, we focus on link Assume that the set of ’s one-hop neighbors is and the set of two-hop neighbors is
Under this neighborhood structure, if changes its power, it will change three links’ virtual SINR, and their virtual rates, i.e., links and
In the algorithm, node randomly select a power level based on its interference to the neighboring links with each feasible power level.
Select decision set: Assume that the number of control slots is and the backoff time generated by the transmitters are
Then, broadcasts an INTENT message at control slot , which is received by and Node will ignore this INTENT message even if it can receive it, because is not within the two-hop range of In control slot , node broadcasts an INTENT message, which is received by node Since both transmitters and receive the INTENT message sent by they are not in the decision set. And the decision set is
Remark: Note that ’s transmit power affects the virtual rate of links and while ’s power affects the virtual rate of link only. Thus, no link’s virtual rate is affected by both of and
Information exchange: Suppose the power level of link has changed in time slot node then has broadcast to all its two-hop neighbors. Node has received this message, which shows that always knows the power level of and
Link selection: Each transmitter only has one link, so if a transmitter is selected to be in the decision set, its outgoing link will be selected.
Critical power computation: Node knows the virtual partial-interference-plus-noise experienced by links
Then, node can estimate its impact on links and when it varies transmit power from 0 to
Fig. 4 illustrates this impact. We can see from the figure that there are 5 critical power levels besides 0 and which are Take critical power level for example, it means that if the power of link is greater than then the virtual SINR of link will be below and link will only be able to use BPSK.
Virtual rates computation: Now, knows the critical power levels and the coding-modulation schemes corresponding to each interval between the critical power levels. Thus, can calculate accordingly, which is shown in Table I. Given these virtual rates, node then samples a power level according to the distribution in equalities (8) and (9).
TABLE I: Critical power level and the resulted virtual rate
In classical Gibbs sampler, the state of each link is updated in a sequential manner. In contrast, the Gibbs sampler used in our algorithm is parallelized and distributed, which leverages the distributed characteristic of wireless networks. In the following lemma, we prove that our algorithm generates a sequence of power configurations which form a Markov chain with some desired stationary density.
For a fixed temperature, i.e., without updating the temperature in the power control algorithm, and fixed queue lengths, the sequence of the power configurations, , generated by the power control algorithm, forms a Markov chain with the stationary density:
where is unknown normalization factor.
The proof is presented Appendix A. ∎
In our algorithm, the virtual powers are updated using an annealed Gibbs sampler. Assuming the queues are fixed, the following theorem states that, with fixed queue length, the power configurations converge to the optimal solution to (5) as goes to infinity. Let and
Let Assume are fixed and is the set of power configurations such that for any
Then given any and starting from any initial power configuration, there is an such that if
The proof is presented in Appendix E. The proof of the theorem follows the idea in . However, in our algorithm, the decision set is randomly generated instead of predetermined, and the Markov chain has a continuous state space instead of a discrete state space, so the convergence of the annealed Gibbs sampling is not guaranteed. The proof therefore is a nontrivial extension. ∎
Remark 1: The theorem requires that the queue lengths are fixed during the annealing, which is the reason the algorithm uses the queue lengths at the beginning of a super time slot for the entire super time slot.
Remark 2: In the algorithm, we replace the interference from non-neighbor nodes with upper bound Therefore, when node changes its transmit power to node to at the end of a super time slot, the actual rate is at least because the actual interference is smaller than that in the virtual rate computation. Further when the neighborhood is chosen to be large enough, i.e., is small, the optimal configuration based on virtual rate is close to the optimal configuration with global interference. But a large neighborhood increases both the computation and communication complexities.
In this section, we use simulations to evaluate the performance of the proposed algorithm, which is SINR-based, with CSMA-based algorithm and Q-CSMA. The CSMA-based algorithm used in the simulation is an approximation of the traditional CSMA/CA with RTS/CTS algorithm. It is implemented as the following. In each time slot, one link is uniformly randomly selected to transmit. Then the links whose receiver is in the carrier sensing range of the selected transmitter are marked and cannot transmit in the time slot. Then another link in the rest of the links is uniformly randomly select to transmit. Repeat this procedure until there is no more link to select. Thought there is no RTS/CTS in the implementation, this algorithm capture the essence of the CSMA algorithm and has similar performance. In the simulations, we assume the channel attenuation over a distance is where the path loss exponent is chosen to be All channels are assumed to be AWGN channels. The transmit power can be continuously adjusted from The rate options for each link are and Mbps, which are the eight rate options available in 802.11g. The system is time-slotted, and each time slot is 1 ms. We assume each packet is of size bytes, i.e., 12 Kbits. So when the link rate is Mbps, packets can be transmitted in one time slot. Each super time slot consists of time slots. is equal to which is the threshold of the channel gain between two neighboring nodes.
Iv-a A Ring Network
Consider a ring network consisting of directed links, as shown in Fig 5. Each node in the network has one transceiver. The length of each link is meters. We assume the carrier sensing range is meters, which is slightly larger than then distance between two nodes that are two-hop away.
The arrival process is the same as the one described in . Namely, at time slot , one packet arrives at the transmitters of links and additionally, with probability one packets arrives at each transmitter. Hence, the overall arrival rate is packets per time slot. In the simulation, we varied from to which corresponds to varying the overall arrival rate from to (packets/time slot).
For each value of we run each simulation for time slots. Figure 6 shows the mean of the sum queue length in the network. It shows that the sum queue length grows unbounded under the CSMA-based algorithm when (i.e., overall arrival rate ). In other words, the network is unstable under CSMA algorithm for On the other hand, our algorithm stabilizes the network for any with a corresponding overall arrival rate equal to Hence, our algorithm increases the throughput by comparing to the CSMA-based algorithm. We can also see that, Q-CSMA, which is throughput optimal under the collision interference model, has similar throughput as the CSMA (around 3). The implementation details of Q-CSMA can be found in .
Iv-B A Random Network
In this simulation, we randomly place links, each with length meters, in a two-dimensional torus. The carrier sensing range is set to be meters, which corresponds to a sensing threshold of dBm. We assume Poisson arrivals for each link, and the arrival rate is the same for all links.
For each arrival rate, the simulations is run for time slots. Fig 7 illustrates the time average value of the total queue length in the network under arrival rates and different rate-control algorithms. We observe that the supportable throughput is packets/slot under the proposed algorithm, packets/slot under the CSMA algorithm. Our algorithm therefore achieves a throughput gain. Comparing to Q-CSMA, we can see that our algorithm has a much smaller queue length and hence has a much smaller delay.
From these simulation results, we observed that our algorithm significantly outperforms the CSMA-based algorithm and Q-CSMA algorithm, which confirms the importance of adapting transmit powers and coding-modulation schemes in wireless network to increase the network throughput.
V Discussion and Conclusion
We remark that there are several important parameters that should be carefully tuned according to the network configuration to optimize the performance.
First, there is a trade-off between complexity and performance in selecting If is equal to zero, all nodes are one-hop neighbors of each other, then the optimal virtual power configuration obtained by our algorithm is the same as optimal power configuration However, the number of one-hop neighbors of each node should be bounded so that the signaling overhead is acceptable in practice. To bound the number of one-hop neighbors, should not be too small. On the other hand, a small is preferred to keep the virtual power configuration to be close to the real power configuration. Therefore, should be carefully determined based on the real network configuration.
Furthermore, the Markov chain converges to the stationary distribution and the probability of being in the optimal power configuration converges to one only when goes to infinity. In practice, we cannot choose to be infinity or too large because the algorithm will response very slowly to queue change and will lead to very bad delay performance. So also should be carefully chosen in practice. Selecting these parameters to optimize the network performance is an important issue of implementing the proposed algorithm in practice. The problem, however, is complicated and requires further investigation, so left as future topics of our research.
In summary, we developed a distributed power control and coding-modulation adaptation algorithm using annealed Gibbs sampling, which achieves throughput optimality in an arbitrary network topology. The power update policy emulates a Gibbs sampler over a Markov chain with a continuous state space. Simulated annealing is exploited in the algorithm to speed up the convergence of the algorithm to the optimal power and coding-modulation configuration. Simulation results demonstrated the superior performance of the proposed algorithm.
Appendix A Proof of Lemma 1
We begin the proof with the following lemma, which states that our algorithm simulates a time homogeneous Markov chain.
For fixed temperature and queue lengths, the power configurations generated by the power control algorithm form a homogeneous Markov chain with state space
Let denote the current power configuration, and denote the power configuration generated by the algorithm.
First, by observing the procedure of the generation of the decision set, it is clear that the decision set is independent of the power configuration Moreover, in the link selection stage, the links are selected based on and Thus, depends on only. Second, for each link the new power is sampled from the density function which is determined by and while is independent of the earlier power configurations than The claim then follows. ∎
Knowing that our algorithm simulates a homogeneous Markov chain, the following lemma gives us the transition kernel density.
Suppose link is selected to update its transmit power, then its power is randomly selected according to the following density:
is a normalization constant independent of
The proof is presented in Appendix B. ∎
It can be easily verified that all the power configurations communicate with the zero power configuration, in which the transmit power of all transmitters are zero. Also,the zero power configuration has a self-loop, which indicates the Markov chain is irreducible and aperiodic and thus ergodic. In the following two lemmas, we will show that conditioning on any link decision set, the detailed balance equations holds, which leads to the conclusion of the lemma.
be the transition kernel probability density, and
For two power configurations if
In other words, if is reachable from in one transition with a link decision set then is reachable from in one transition with the same link decision set.
The proof is presented in Appendix C. ∎
For any link in the link decision set if
The proof is presented in Appendix D. ∎
Appendix B Proof of Lemma 4
First, we arrange the critical power levels of
in ascending order, denoted by
Recall that the power level of link is generated using the following procedure:
Node first selects a power interval with following probability:
Given the interval is selected, then randomly select the power level according to the following probability density function(pdf):
For any power level let us consider the probability density. it must in an interval for some Hence, is selected according to the following density:
We then need to show when