MeasurementAdaptive Cellular Random Access Protocols ^{†}^{†}thanks: This work was supported by AlcatelLucent Bell Labs, Stuttgart, Germany.
Abstract
This work considers a singlecell random access channel (RACH) in cellular wireless networks. Communications over RACH take place when users try to connect to a base station during a handover or when establishing a new connection. We approach the problem of optimal coordination of user actions, taking into account a dynamic environment (channel fading, mobility, etc.). Within the framework of SelfOrganizing Networks (SONs), the system should selfadapt to such environments without human intervention. To do so certain information should be gathered at the base station. For the performance improvement of the RACH procedure, we propose protocols which exploit information from measurements and user reports in order to estimate current values of the system unknowns and broadcast global actionrelated values to all users. The protocols suggest an optimal pair of user actions (transmission power and backoff probability) found by minimizing the drift of a certain function. Numerical results illustrate the great performance benefits at a very low or even zero cost in power expenditure and delay, as well as the fast adaptability of the protocols to envoronment changes.
Keywords:
Random Access Channel SelfOrganizing Network (SON) Measurements Collision Resolution Drift Minimization Power Control∎
1 Introduction
Random multiple access schemes have traditionally played an important role in wireless communication systems. Their use has been established especially in cases of bursty source traffic, where a multiplicity of users requires access from a central receiver. Starting with the ALOHA protocol Abramson (), several modifications have been suggested in the years to come aiming at performance improvement EphreTutor (). A very common application is in wireless LANs, such as the IEEE 802.11 protocol (see BianchiRACH (), GuptaRACHINFOCOM (), SharmaINFOCOM06 () and references therein). The random access channel (RACH) is also included in the 3rd Generation Partnership Project (3GPP) as an important element within the Long Term Evolution (LTE) of cellular systems 3gppRACH (), 3gppson (), GPPTS36321 ().
In the case of wireless cellular networks, a very limited frequency resource is reserved for the cases when a user requests for access from a base station (BS) or in order to be synchronized for uplink/downlink data transmission. RACH communications further occur during the handover phase Kiki11 (), because of user mobility, or when a user is (re)initiating some new service. RACH channel can be used as well during the load balancing procedure GiovWSA12 (), when celledge users are pushed to migrate to a neighboring BS after modification of the cell individual offset.
Due to limited resources, connection failure can occur in cases when the system is not well adapted to the incoming traffic. Consider for example large spaces in cities where occasionaly a vast amount of requests for service can be demanded, although normaly the system is not heavily loaded (e.g. metro stations, market streets, stadiums, city squares, areas close to concert and conference halls etc.). In such places, it is very common that the system fails to support the service for all users and one of the reasons can be high collision rate in the RACH channel. It is thus necessary, within the context of Self Organizing Networks (SON) 3gppson (), OsterboRACH () that the system can adapt to abrupt environmental changes that influence its functionality. Thus the RACH selfoptmization problem is identified as an important case in the LTE standardization process (3gppson, , paragraph 4.7).
Unfortunately, in all such cases, the cellular system has almost zero userspecific information. Each BS can however broadcast certain information with cellspecific access details SelfOpt3G (), which allow the users to adapt their operation. Furthermore, carrier sensing as understood in the 802.11 is here not possible, which provides limitations to the design of high performance protocols. This is because, the possibility for a user to sense whether the channel is idle or not, is not provided and collision events cannot be avoided.
The procedure is called random access, due to the fact that the users access the channel in a random fashion. In the ALOHA case, when more than one user transmit simultaneously and their signals are detected we say that a collision occurs and all efforts are considered unsuccessful. LTE standardization, instead, provides the possibility for each user to randomly choose over a common pool of orthogonal frequencies 3gppRACH () and a collision takes place when at least two users make the same choice during the same transmission interval. After a failure, each source enters a backoff mode. The period of user silence is usually chosen having an exponential distribution but other possibilities can be used when such choice is adapted dynamically. This backoff time can generally be modeled in the slotted case by a per slot probability of transmission, less than . Using this technique, an increase in throughput is achieved at the cost of additional delay. Furthermore, since the detection or not of a user signal is also critical for the success, an important parameter is the transmission power of each user as well.
In short, the access (backoff) probability and the signal power are the two user actions. An interesting idea to improve the decision making is to make certain global information of the system state available by broadcasting it from the base station. This is compatible with LTE standards where other type of information is already considered as globally known 3gppRACH (). The information should represent the current system situation, so that users may adapt their actions dynamically. In this way the delaythroughput tradeoff can be enhanced. The cost is certain signaling and computations for the updates at the BS side. Furthermore, the BS should have a way to gather relevant empirical information from its environment, related to the RACH functionality.
Based on the above idea, the current work suggests a dynamically adaptive RACH protocol for the cellular systems focused on LTE design. Empirical information is gathered through measurements and user reports. After certain processing at the BS side global system parameters are broadcast to users who require access. The protocol suggested, which is based on adaptation of the system to changes in the environment, further guarantees nearoptimal performance related to a certain performance metric.
1.1 Related Literature
Bianchi BianchiRACH () has been the first to provide a precise performance analysis for a random access protocol, which uses exponential backoff times. His approach considers a saturated system model, where the number of users is kept fixed to and all have a packet to send at each time slot. The results are based on the key approximation that the collision probability of a packet transmitted is constant and independent, which decouples the evolution of the system to 1dimensional Markov Chains.
A different approach has been suggested by Sharma et al. SharmaINFOCOM06 (), where more general backoff strategies (generalized geometric) are considered for the IEEE 802.11 protocol in order to take service differentiation into account. One of the major differences is that the system state is described by the current number of users per effort, while the collision probability is not independent per user.
First suggestions for dynamically controlling multiple access protocols can be found in Hajek and van Loon HajekRACH () as well as Lam and Kleinrock LamKlein75 (). More recently Markov Decision Processes (MDPs) have been used in delAngel04 () to derive optimal power and backoff policies for a set of backlogged users in slotted ALOHA random access systems. Cases of unknown user number have also been taken into account.
Gupta et al GuptaRACHINFOCOM () have recently suggested a dynamic backoff adaptation mechanism, where contention is regulated by broadcasting a so called contention level to the users. This is similar to the idea used in our approach. Works of particular interest are also those of Liu et al Liu09 () and Cheung et al Cheung10 () which use the framework of utilityoptimization for the optimal choice of transmission probabilities.
Channelaware scheduling approaches in conjunction with random access mechanisms (which do not find application here due to the lack of such information in the system) include DimicTutor (), TongTutor (), and more recently BorstRandomAccNEW ().
How random access works in the 3GPPLTE systems is thoroughly described in SelfOpt3G (), where certain suggestions are presented, related to a selforganizing mechanism with information exchange between users and the Base Station. Investigations on the RACH power control include LeeRACH () and references therein, whereas an analytical framework for RACH modeling and optimization is given in YilmazRACH ().
Finally, rather interesting for the CSMA/CA case is the dynamic adaptation mechanism suggested in IdleSense () where users adapt their time window based on measurements and estimation of the average number of idle time slots of the random access channel. It involves an AIMD (Additive Increase Multiplicative Decrease) rule for the updates. Unfortunately, such a technique cannot be directly applied to the cellular system due to the unavailability of the sensing mechanism, it can however give ideas for application of a similar mechanism for the power updates.
1.2 Contributions and Outline
We investigate a saturated system model, where a number of users are always present within a wireless cell and try to gain access to the Base Station. An effort is successful when the user transmits a certain sequence, which is detected at the Base Station and at the same time no collision occurs. The event of collision will happen when the transmitted sequence of another user is also detected. Furthermore, LTE standards allow for orthogonal sequences randomly chosen by the users, so that even when two user signals are detected, access to both may be granted.
In our analysis the missdetection probability and collision probability are left as unknown variables. However, higher power increases the chances for detection and reduces collision probability, whereas use of access (otherwise backoff) probabilities reduces the collision events. Transmission power and access probability are the user action pair.
After description of the action space and state space, the transition probabilities are given and the evolution of the system is described by a Markov Chain. Furthermore, the event of dropping, when the users exhaust the maximum number of efforts allowed, plays an important role. Unfortunately, due to the unknown expression for the success probability no steadystate analysis is possible. The above are analytically presented in Section 2.
What we can do however, is to choose the actions myopically optimal, in the sense that they optimize the expected change in one timeslot for some function of the state space. For this we introduce in our analysis the drift of a delayrelated function. To motivate further our formulation, it is shown in the Appendix how the solution of the drift minimization problem is related to the solution of an ideal Markov Decision Problem for optimal performance in the steadystate. Our problem formulation is found in Section 3.
To solve the problem online a protocol is introduced. Its steps are presented in Section 4. The BS collects measurements as well as user reports to estimate the unknown probabilities (missdetection, contention, success) at the Base Station side, as well as the current number of users, which is actually unknown in a real system. After solution of an optimization problem and a closeloop control problem, the BS broadcasts two values, the current contention level and the current power transmission level, so that the users can update their action pair.
2 System Model
2.1 General Description
We consider an arbitrary but fixed total number of users labeled by trying to randomly obtain access to a cell Base Station (BS) over the wireless channel. The time is slotted, with each slot interval normalized to and indexed by . At each time slot all users belonging to the user set have the possibility to access the channel by transmitting a preamble sequence (as specified in the LTE standards). There are two criteria that determine the success of an attempt.

The SignaltoNoise Ratio (SNR) at the BS exceeds a predefined detection threshold . If the SNR is below the threshold, we assume that a missdetection occurs and the user has to retry. The Detection Miss Probability (DMP) can be written as the probability of an outage event
(1) where is the chosen transmission power and the probability is taken over the random channel quantity denoted by and is i.i.d. over time . In general we will consider that the BS does not approximate somehow the expression for outage. This is reasonable since the information over the user positions and the exact fading statistics is not known a priori.

No collision of transmitted signals occurs. Typically in the slotted ALOHA protocol Abramson (), when more than one user attempts to access the channel during the same time slot a collision occurs and all affected users have to repeat the effort. In more recent wireless protocols, such as those suggested in LTE standards GPPTS36321 (), a pool of orthogonal sequences (e.g. ZadoffChu) is made available to all users. Each user chooses one sequence from this set randomly (uniform distribution) and the probability of collision can be made less than when two users transmit simultaneously.
In our model, the probability of collision is conditional on the transmission and the detection of signals at the BS side. That is, a user may collide only if he transmits at time slot and his signal is detected. Assuming that users transmit at time slot with transmission probability vector and outof (we write ) are detected, the overall Collision Probability (CP)  the probability that at least one collision occurs  is an increasing function of both and
(2) As in the case of the DMP we consider that the base station does not have an exact closed form expression to calculate the CP and the above quantity is in general unknown.
2.2 Action Space
There are two actions that user can take for transmission at time slot .

The choice of the transmission power level , which influences the detection of the transmitted signal at the BS, as shown in (1) and eventually the collision probability (through the number of detections ). In general exhibits a monotone decreasing behavior with respect to power.

The choice of the access (or transmission) probability per user, at a given slot . This influences the number of simultaneously transmitting users in the cell and therefore directly affects the collision probability in (2). The backoff probability simply equals .
The set of actions for the entire system of users at is denoted by the dimensional vector . The action space per timeslot is denoted by and is the Cartesian product , where is a given individual user power constraint per slot. Furthermore, .
Until the end of the subsection, we provide a discussion on the influence of choice for the backoff probability. In the definition (2) no backoff action is taken, , and all users transmit simultaneously. On the other hand, assigning to some users, displaces the transmissions in time and the effect of collision is mitigated. Since less than users simultaneously compete for the access of the medium in some slot , the collision probability is reduced. This can also be shown analytically.
The overall collision probability of users present within the cell, with access probability length vector , and exactly users detected, equals
(3) 
where is the probability that  given a probability vector  exactly outof users in the cell transmit. The equality follows from the total probability theorem, since the union of events transmissions exhaust the sample space. The transmission probability of users equals
where the summation over is taken over all possible combinations (sampling without replacement) of users transmitting and users remaining silent, is the index of user belonging to combination that transmits and is the index for the user that does not transmit.
Proposition 1
Given (the inequality means that for at least one ) and exactly detections, we have that
(4) 
Proof.
: The events exhaust the sample space and we have that their probability sum equals . Furthermore, for it holds since there cannot be more detections than transmissions. The higher the number of transmissions, the higher the collision probability, which means , and the inequality is strict for . From (3) we have
which concludes the proof. ∎
2.3 Success Probability, Failure Event and Dropping
From the above, success of a transmission is an event which occurs when (i) a user transmits, (ii) the user signal is detected and (iii) no collision occurs. In the use of orthogonal sequences/preambles, it suffices that no two users sharing the same sequence collide. In general, conditioned that a user transmits, the Success Probability (SP) equals
(5) 
Observe, that the success probability of a single user does not depend only on his own action set , but also on the choices of access probabilities of the other users, as well as the number of detected users . The latter is further dependent on the transmission power chosen for , so we can instead write
(6) 
In the case of an unsuccessful effort the user may retry. Each user is constrained to at most access efforts and the efforts are indexed by . After unsuccessful efforts the user is considered discarded and replaced by a newcoming one, so that the total user number in the system always remains equal to . The same holds when a user leaves the system after success. Therefore, we say that the system is saturated. The number of users at effort in time slot is denoted by and from the above it follows that
(7) 
We occasionaly write in the following that a user at effort belongs to user class .
2.4 System States and Transition Probabilities
We define the state of user at slot as the current transmission effort , whereas the system state as the dimensional vector
(8) 
Altogether, there are different user states and different system states (e.g for a cell with users and maximum efforts, the number is approximately million). The entire state space is denoted by . It is easy to verify that the system state forms an dimensional Markov chain.
We group the transitions for each user into (a) returning to state in case of transmission and success, (b) moving to the next effort in case of transmission and failure and (c) backingoff and remaining in the same state. The expressions for the transition probabilities are given below. (Dependence of the functions on other parameters except the time index is omitted for brevity of presentation.)

For :
(9) (10) (11) 
For the user boundary state :
(12) (13) A user in state will either backoff, in which case he remains in the same state, or transmit. When a user transmits, he will either succeed or fail. In both cases the next state is set to 1, the user is removed from the system and is replaced by a new one so that the total number is always equal to . The transition probabilities in (12)(13) for coincide with those for , given by (9)(11) when . In other words, to keep the system saturated, the Markov Chain evolves as if transmission at state always results in success.
This is why, it is further important for the analysis to specify the user Dropping Probability (DP)
(14) 
If the exact expressions for the DMP and CP were available, it would be possible to calculate the steady state probabilities of the system, by forming the transition probability matrix and using the PerronFrobenius theory (NonnegativeMatrix, , Ch. 2 and 8). Since the number of states is finite, and for each user the probabilities (9)(11) and (12)(13) sum up to (stochastic matrix), a steady state with probability sum equal to always exists, although certain states may be transient and have zero probability.
3 Problem Statement as Drift Minimization
Since the exact expressions for the detection miss probability as well as contention probability are unknown (hence the success probability , which appears in (9) and (10)), it is not possible to use the standard steadystate analysis as followed in TakKlein85 (), Maglaris87 (), Proutiere08 (), Paschos07 (), Klein75b () and Liu09 () (among others) to derive longterm performance measures and optimize the system. Even if this would be possible however, the solution of a system of such an immense number of variables would bring difficulties (remember the number of 10 million variables for and ). The same problems are met in a Markov Decision Problem (MDP) formulation, as followed e.g. in LamKlein75 () and delAngel04 ().
Furthermore, in a realistic setting, we would like to propose a protocol, which takes into consideration the fact that within the wireless cell, users appear and leave the system after a while, whereas the fading situation changes unpredictably. These two factors greatly influence the missdetection and collision probabilities, which do not remain fixed until infinity, but exhibit large fluctuations over time. This falls within the concept of SON’s which should selfadapte and selfoptimize the wireless system parameters as a reaction to such unpredictable changes from outside without human intervention.
For the above reasons we make use of the notion of drift for the Markov Chain under study, in order to achieve an improvement in the system performance by appropriate choice of actions. The idea of drift is commonly used in the literature of stability of systems with infinite states TassQue93 (), TassMultihop (), NeelySat (), NeelyRouting (). In such cases, if we can find, for a given positive Lyapunov function, an action policy which keeps the drift negative for the entire state space  except possibly for some finite subspace  the system is guaranteed to remain stable. This comes from direct application of Foster’s theorem (see (Asmussen, , Prop. 5.3(ii))). Intuitively the negative drift gives the function of states a tendency to decrease in expectation at each step, as long as it is outside the aforementioned subspace, so that in the long run the value a state can take will not be unbounded (and the stability is guaranteed). In our case the state space is finite due to the finiteness of . However, since the amount of users that exceed efforts are eventually dropped, stability of the system refers to keeping the number of dropped users finite. (Alternative application of the drift minimization to a problem with and no dropping does not change much the policy and results).
The drift equals per definition, the expected change in the Lyapunov function from to . By choosing an appropriate nonnegative function of the system state related to some performance criterion, we can choose actions that optimize performance at each timeslot. Since it is impossible to know how the system will evolve in future slots, and since expressions for DMP and CP are not available, the best thing we can do is to provide a onestep lookahead (myopic) policy for the system, given its current state and measurements performed on time , which estimate unknown parameters. Specifically, given that the system state at is , the drift is defined as
(15) 
and is also a function of the action set , since the actions control the system state transition probabilities .
The function to be used is the sum of user states and is linear. It can be rewritten as the sum of cardinalities of users at a state, weighted by their effort index.
(16) 
A user who is currently at a higher effort, contributes more to the function, than users at lower ones. By minimizing the drift of such function we wish to choose appropriate actions in order to have success with as few efforts as possible. This has following objectives:

keep a good tradeoff between power consumption and delay until success per user

diminish the proportion of users who are dropped

maximize a notion of total system throughput
To understand the last point, observe that each user contributes a ratio to the total system throughput if efforts are required for success and contributes nothing if the user is dropped. Consider now as a single virtual user, the set of users in the network. By use of the RenewalReward theorem GiovSCC08 (), the longterm throughput of such a virtual user (considering only number of efforts and not the total number of timeslots required including user silence slots) will be the ratio . Alternative Lyapunov function could change the objective of the minimization, giving emphasis to total delay or power consumption and can be understood as alternative formulations of the same general problem and solution methodology.
Let us consider statedependent, rather than userdependent actions, in the sense that all users who are at class in slot should make the same choice for transmission power and backoff. The specific drift expression can now be derived to yield
(17)  
The drift minimization problem at each time slot is
(18) 
A further motivation to pose the problem as a drift minimization is provided in the Appendix. It is shown that (18) is a myopic solution of an MDP with objective the minimization of the expected Lyaponov function at the steadystate (for ). For the formulation and solution of the MDP, the expression for , should be available and the channel/user statistics should remain unchanged over the entire time horizon.
What is needed to solve the above problem per slot? It follows from (17) that the following information should be available at the BS side:

The cardinality of users at each effort .

The current value of at each .

The current value of .
Using 2. and 3. and the product in (5) the actual value of can be obtained. Although the BS does not know these values it may estimate the variables and with it approximate the objective function, using measurements related to channel and service quality, as well as information reported directly by the user set. The goal is to use these estimates for optimization, in order to achieve significant performance gains, while keeping an additional overhead of exchanged information as small as possible.
In this way, a sequence of problems with different numbers of users, contention and missdetection probabilities can be solved over time, which help the cell to follow and adapt to dynamic unpredictable changes. The steps of the proposed adaptive protocol are summarized in Table 1.
4 Five Steps of the Protocol
Before proceeding to the algorithm, we first discuss over the action pair of access probabilities and transmission powers. Considering the access probabilities, we adopt the approach in GuptaRACHINFOCOM () (similar functions are also found in Liu09 () and references therein), with per effort probability given by
(19) 
Here and hereafter, is called contention level and is some fixed function of the transmission effort. In this way, a simple variable can simultaneously define the entire set of transmission probabilities. By choosing to be monotone increasing in , priority is given to users with higher efforts, while such users obtain lower priorities when is strictly monotone decreasing. Typical backoff protocols follow the exponential rule, which reduces by half the probability of accessing the channel after each failure, so in this case and . Other possible choice could be , (in this work and the simulations to follow the case is mostly used). Exponents will lead to an overly conservative system with large delays for users in higher states, whereas tends to treat users of all classes with the same priority. In the following, the expression in (19) will sometimes be replaced by and the constraint is taken into account in the constraint set of the minimization problem.
We consider, furthermore, the transmission power to vary per effort as a ramping function. This approach is often considered in practice (for related approaches, the reader is referred to SelfOpt3G () and references therein). The power level for the first effort is given by and for all efforts by the expression
(20) 
where is the ramping step with a fixed (tunable) value. Thus, analogously to the case of the backoff probabilities, the vector of power actions can be defined by appropriate choice of the power level per time slot.
4.1 Step 1: Measurements and User Reports
When users attempt to randomly access the channel, we assume that the BS counts the overall number of detected user efforts, as well as the overall number of successful efforts. Given an observation window of length , both the quantities depend on the time interval and are denoted by and respectively. Furthermore, after every successful effort, the users are assumed to report to the BS, the total number of trials required to get access. In this way, the BS can keep track of the number of successes at effort , within the observation window, denoted by . The reports over the success state also provide information over the overall number of transmissions of users being at some state . As an example, if within the observation period two users report success at effort and respectively, the BS can estimate the number of transmissions at state by , at by and at state by , without considering users that have yet not declared success, or are dropped. We denote these estimates by and their sum, which equals approximately the number of access efforts within the observation window, by . Altogether, the set of gathered empirical information, updated per time slot, is represented by
(21) 
4.2 Step 2: Estimation of Unknowns in the Objective function
Using the above counters, we can now approximate the unknowns in the expression (17) that are briefly discussed in points 1.  3. in the previous Section.
As far as the unknowns in 2. and 3. are concerned, the actual overall contention probability and per effort success probability in (5), can be estimated by contention and success rates, an idea which has already appeared in SelfOpt3G (). Observe that the additional information about the per effort missdetection probability cannot be deduced from the above measurements. What can be calculated, instead, is an overall rate of missdetection (DMR), without differentiating between efforts, which we denote by .
(22)  
(23)  
(24) 
Regarding the number of users currently within the cell (discussed in 1.) and their estimation, we proceed as follows. Instead of attempting to find integer values, we consider arrival rates. As the total arrival rate of users we consider the ratio , which is the time dependent ratio of accepted users, divided by the observation window. The above is used under the assumption that only a very small fraction of the users are dropped throughout the process, so that almost all users appearing within the cell, will eventually have at some point a success. Taking dropped users into account requires an additive correcting term that may be deduced from empirical observations.
The window is considered long enough, so that the resulting success rates per state, in (23), approach the actual success probability per effort. These can replace the entries in the onestep transition probability matrix in equations (9)(11) and (12)(13). The steady state probability distribution is found by solving the system , where is the row vector of the unknown probabilities for the states with and is the transition probability matrix. The solution equals
(25)  
(26) 
The ratios of the unknown backoff probabilities are involved in the expression above. From the previous discussion , which is known since the function is chosen a priori. With these observations and definitions at hand, we can estimate the user arrivals per effort according to
4.3 Step 3: Solving the Problem
Once step 2 is performed, we can formulate the objective function to approximately solve problem (18) and with it find the optimal actions per time slot. To this end, we break down the problem into two subproblems and propose two subalgorithms based on the measurements and estimated quantities described above.
Backoff Probability Problem: The objective function at the base station is estimated by
(28) 
where the success probability is substituted by the success rate in (23) and the average user number by the expression in (27). As long as such estimates are close to the actual values and are considered reliable, the BS can solve a problem with parameters adapted to the changing environment.
When the expression in brackets above is positive, the objective function is convex and decreasing in the contention level variable (behaves as ). When is negative, the objective is concave and increasing in (behaves as ). Due to the monotonicity and concavity/convexity, the optimization will have as a result either maximum or minimum value of depending on the sign of the term inside the square brackets.
In the following we provide the boundary values and of the domain of . The lower bound on follows from the fact that all access probabilities are less than or equal to :
(29) 
To obtain an upper bound, we further provide a constraint on the probability of a time slot being idle (no user transmits). This probability is less than or equal to , which is a design factor for the system.
(30) 
The left handside is increasing with , thus the inequality provides an upper bound on . If we solve (4.3) for equality, we then derive the value of . Notice furthermore that, all values of within the interval are feasible solutions of the contention level.
Proposition 2
Considering the problem of minimizing in (28) subject to the upper and lower bound constraints on , the following necessary and sufficient optimality conditions hold:

if then the optimal contention level equals and is found by solving
(31) 
if then the optimal contention level equals
(32)
Power Control Problem: In order to identify optimal transmission levels, one could proceed along similar lines as above, to formulate an optimization problem, given the backoff probabilities and the contention rates from (22). In order to determine the objective function based on (17), which is denoted by , the closed form expression for the detectionmiss probability as a function of power may be necessary. It is however unlikely that the channel’s fading behavior in practical systems can be accurately represented by a closedform expression, especially since in the random access cellular system the user position is not known to the BS.
A different approach  which is adopted here  is to use a MultiplicativeIncreaseAdditiveDecrease (MIAD) control rule, as in the case of congestion control protocols in TCP CongestionRef (). In this way, the BS reacts to the change of the estimated DMR stepwise, by increasing or decreasing the power level per time slot, depending on the current value . We set two levels of action, a high detectionmiss level and a low one . The control loop then works as follows: When is exceeded, the power level is increased by multiplication with a tunable factor . This action increases considerably the transmission power since missdetection is highly nondesirable. When the ratio falls under the low level , which is considered satisfactory for the system performance, the power is reduced in a conservative way, to reduce the energy consumption on the mobile devices, by subtracting a constant tunable amount of . For instance can be set equal to the ramping step in (20). The control loop is then described by the power updates
(33) 
Obviously, updates on the pereffort ramping steps or userspecific power control could be much more beneficial instead of the update in the global power level . Furthermore, it is obvious that by varying globally, power consumption will increase not only for users in higher efforts but also for those in their first effort, which may not be necessary. However, there are certain difficulties in providing a different type of feedback. Most importantly, there is no user channel state information available at the BS and channel adaptation is impossible. Furthermore, based on the possible approximations that  given the measurements and the reports  are suggested, only a global missdetection rate can be estimated in (24) and no statespecific or userspecific rates (say ). We cannot approximate, in other words, the rate of missdetection for a user at different states and as a result we cannot suggest different statedependent power levels. Finally, statedependent power control would increase considerably the feedback information broadcast to all users. For all the above reasons, the suggestion of the MIAD rule was considered more appropriate.
4.4 Step 4 and 5: Broadcast of Information to the Users and Action Calculation
The last two steps of the proposed algorithm involve the broadcasting of the actionrelated information to the users and the choice of appropriate actions by them. The broadcast information includes the pair consisting of the contention level and the power level
(34) 
Let us assume that the expressions in (19) and (20) for the success probability and the power level per effort are known a priori to the mobile stations. Since each user is aware of its current individual state , calculation of its own action pair is possible, according to
(35) 
Note that if the required power and access functions ( and the ramping step ) is not available at the mobiles, the BS could broadcast the entire vector of computed transmission powers and access probabilities to the users so that they choose the actions according to their current effort.
A remark considering implementation issues of such protocols is that the updates of these two levels are not expected to take place very frequently, but rather only at the rate of estimated change of user traffic and fading conditions. Furthermore, user reports and broadcast feedback from the BS is already suggested in standardization reports, so that the proposed protocol complies fully with the existing standardization literature 3gppRACH (), 3gppson (), GPPTS36321 (), without introducing additional protocol information.
5 Numerical results
5.1 Description of the simulations setting
The proposed algorithm has been implemented in a single cell scenario. The users are randomly positioned, with a 2D uniform distribution and the algorithm is evaluated for the cases of [users/time slot] present in the cell. Considering the transmission scenario, each user randomly chooses at each attempt one sequence, out of a pool of orthogonal sequences, and transmits with a chosen backoff probability and transmission power. The number is used for simulation purposes, whereas the actual number suggested in the LTE literature equals ; however not all users have access to the entire pool of sequences (see 3gppRACH ()) since the sequence allocation procedure is more complicated than the simple uniform choice we use here.
The signal experiences path loss due to the userBS distance. Fast fading is initially not modeled (this will be considered in the second part of the Section for the power consumption evaluation) but the channel is considered AWGN with noise mean equal to dBm. We have to note that in case fastfading were also implemented, a further randomness in the channel would affect the signal detection and the protocol performance. To keep things simple, we consider first only the randomness of user positioning which affects the slowfading coefficients  also unknowns during the procedure. The evaluation of the protocol’s performance will not change much by adding more randomness factors.
An effort is successful when among the detected sequences there exists no pair that collides, in the sense that no two detected users choose the same sequence for transmission. A user is dropped when the effort fails at the maximum access effort . After a success or an event of dropping, users are removed from the waitingfortransmission list, and the same number of newly arriving users are added, each given a random position on the plane.
Power and access probability for the users are computed per slot equal to the action pair in (35), for . The choice of exponent is not conservative (whereas a higher exponent would be) while at the same time it takes class differentiation into account. Important is to notice that the expression of the function greatly affects the delay. On the other hand, the delay can be controlled by the parameter which is systemoperatordependent and tunes the expected idle period. The set of values for the parameters of the system simulation are summarized in Table 2.
Several factors for the protocol design have been left open for choice. One of them, as mentioned already, has been the desired idle probability . The higher factor is, the more the delay suffered by the system but the higher the benefits in dropping rate and power consumption are. Other important parameters are the steps and bounds , of the MIAD rule, the access function and the adaptive window length , which defines how fast should the protocol adjust to environmental changes. A summary of these tunable factors and how they are chosen within the simulation setting under consideration is provided in Table 3.
5.2 Comparison to a Fixed ”open loop” Power Fixed Backoff protocol
The suggested algorithm is compared to a scenario, where access probabilities and target power are held fixed, while the ramping step for the transmission power is predifined and same for all efforts. The fixed scenario is in other words an ”openloop” control scheme, with predefined constant . The choice for the fixed backoff probability in the comparison scenario, equals and is such that the average occurance of an idle slot is less than , hence the channel is kept busy with user efforts for access during most of the time . In this sense, the comparison between the adaptiveprotocol suggested and a fixed protocol is more fair for a tunable factor of or less. How the average idle probability changes between and the fixed case can be seen in Fig. 1. We refer the reader to the Parameter Table 2 for the actual values used throughout these simulations. The above fixed scenario is denoted by (FPFB) for Fixed Power Fixed Backoff. Two types of protocols are used for performance comparison:

Fixed Power Dynamic Backoff (FPFB) protocols. In this case the ”open loop” power control of the protocol is the same as in the fixed scenario FPFB case. The backoff mechanism adapts to measurements as suggested in the protocol description of this work (Paragraph 4.3, Backoff Probability Problem).

Dynamic Power Dynamic Backoff (DPDB) protocols. In this case both backoff and power are adapted as the protocol suggests in Paragraph 4.3. The backoff comes from the solution of the drift minimization problem, while the target power is adapted according to the MIAD rule.
5.3 Performance Evaluation: Lyapunov Function and Number of Efforts
The performance of the scheme and its comparison to the fixed scenario FPFB is initially illustrated in the plots of the performance metric in Fig.2 and the plots of the average number of access efforts until success in Fig.3. The two figures show a close relation to each other, due to the choice of the specific Lyapunov function . Since was chosen as the sum of user efforts, lower values translate into better performance for the protocol. In all six curves, our protocol outperforms the FPFB scenario in the metric chosen as well as in the average number of user efforts. Furthermore, all DPDB cases schow improved performance compared to FPDB, given a certain value of the parameter . The higher the value of tunable factor , the better the performance and the less the averge efforts required up to packet reception.
5.4 Performance Evaluation: Delay, Power Consumption and Dropping Rate
The three most important performance measures in random access that can illustrate the improvements of the suggested protocol are the total delay suffered by a packet until success (including backoff slots), the total transmission power used until success as well as the percentage of users dropped because the maximum number of efforts is exceeded. These are shown in Fig.4(a), 4(b), 5(a), 5(b) and 6(a), 6(b) respectively, for (a) the FPDB case and (b) the DPDB case.
From the plots, it is illustrated how an increase of the parameter influences positively power consumption and dropping rate at the cost of delay. Furthermore, the DPDB schemes perform better than the FPDB schemes in terms of delay and dropping, but have a cost in power consumption. Altogether, the performance of the protocol is tunable, to the requirements of the service provider. If the delay is not an issue, power can be considerably saved and the number of users dropped is reduced. As long as delay becomes an issue, transmission power can still be saved by using only the FPDB protocols. The dropping rate is also improved in such a case. Dynamic backoff generally allows the system to remain stable  in the sense that the rate of dropped users does not tend to ”explode”  for a higher value of . The behavior of this measure also improves for higher , which is reasonable since allowing a higher idle probability, distributes the transmissions of users among a larger number of timeslots.
A more detailed comparison of the schemes is given in the following figures. Specifically, Fig.7(a) and Fig.7(b) illustrate the beneficial use of the MIAD power control for the detection miss ratio, which leads to a drastic reduction of the average number of missdetected signals in the system for DPDB protocols. Obviously the missdetection curves for FPDB are similar to the FPFB case, since no power control is applied. Furthermore, considering the contention ratio CR, both Fig.8(a) and Fig.8(b) show benefits compared to the fixed FPFR case. Interestingly, the DPDB cases are slightly worse than the FPDB. This is because a higher number is detected for the same window size , so that the CR calculated as in (22) appears higher.
5.5 Protocol adaptation to channel fluctuations and deep fades
In the current subsection, we further illustrate the performance of our protocol  which operates with parameters given in Table 3  for a scenario with fluctuations and abrupt changes of the fading conditions. Such investigation shows how fast and with which cost in power expenditure can the protocol adapt to environmental changes. Specifically, we use a factor to multiply the longterm fading of each user. Initially the factor has an expectation and its value fluctuates uniformly within the interval . After a certain timeinterval we initiate a sudden deterioration of the channel to an average of , which returns to after some time. The realization of such fading scenario for a given user is presented in Fig. 9(a).
Very important here is to show how the protocol performs over time and adapts to the changes. Compared to the fixed power scenario, our suggested protocol can react very fast to the changes by an increase in power consumption during the period of the deep fade, which keeps the DMR always within the defined interval . This can be observed in Fig.9(b) and Fig.9(c).
6 Conclusions
We have suggested a dynamically adaptive protocol which updates the user access probabilities and transmission powers in cellular random access communications for LTE systems. The protocol is based on measurements and user reports at the base station side, which allow for an estimation of the number of users present within the cell, as well as the quantities of detectionmiss and contention probability. The protocol updates take place per time slot in a myopic fashion. By solving a drift minimization problem for the contention level and using closed loop updates for the transmission power level by a MIAD rule, the BS coordinates the actions chosen by the users, by broadcasting the pair .
The algorithmic steps, together with the methodology of the drift minimization for a certain measure of interest referring to the steady state, provide a general suggestion to treat problems of selforganization in wireless networks. Considering the specific scheme, a large variation of algorithms can be extracted, by choosing e.g. some different state function for the performance measure, or by introducing other kinds of user reports, which may provide more information to the central receiver, at the cost of increase in signaling. Furthermore, a larger action set can definitely provide a higher performance, compared to the proposed one  which introduces two possible values for the contention level (high/low) and two actions for the power level (increase/decrease). Even in this scheme however, which is characterized by an “economy” of signaling and information exchange, the results  as illustrated by numerical examples  are extremely beneficial, especially as the user number in the cell increases.
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Appendix  Relation between the Drift Minimization and a Markov Decision Problem Solution
We begin this section by considering an ideal setting, meaning that all expressions are known and the system is fully controllable by the choice of actions. Let be a nonnegative function of the system state and let be a performance metric related to the steady state reached when , if the initial state is . The metric is a function of the entire set of actions
(36) 
If the actions are chosen per timeslot from the set , the following general MDP can be posed:
(37) 
Proposition 3
The MDP in (37) can be solved using the dynamic programming tools. The optimal solution satisfies Bellman’s equation Puterman ()
(38) 
for the costtogo function , where is the possible state at the next time slot, while the transition probabilities are functions of the actions chosen. The solution is statedependent, meaning that the optimal actions depend on the system state and not on time.
Corollary 1
Proof of Proposition 3
We first need the following lemma
Lemma 1
The performance measure can be written as an infinite sum of expected drifts over the discrete time axis, given the initial state
(39) 
Proof.
: Let be the information over the system realizations up to slot . Obviously (formally we call a filtration and is a subalgebra of ) and the tower property for expectations (Williams, , p.88) holds. Hence,
and by repeating the process for and taking the limits for we reach the result. ∎
Now we can continue with the proof of the the Proposition. Consider the series in (39) up to a finite horizon and denote the related sum by . Then the expected drift term for some equals
It can be observed that , which can be controlled by the actions appear in all summands of , for and not for . Following this observation, the optimal choice of actions are found by solving , the costtogo at .
The costtogo can be verified to satisfy the recursion, :
The expression holds as well, when we let the horizon . Thus taking results in (38).
Tables
STEP 1  Gather empirical information at the BS. 

STEP 2  Estimate unknown factors (see 1.  3. above). 
STEP 3  Solve the resulting optimization problem in (18). 
STEP 4  Broadcast actionrelated information . 
STEP 5  Calculate at the user side the required actions, based on . 
Parameters  Value 

Wireless Network  Single cell 
User distribution  Uniform within cell 
Number of users in cell  
Sequence pool size  
Fixed Tx Power  mW 
Power ramping step  mW 
Maximum Tx Power  mW 
Path loss  dB 
Noise  dBm 
SNR threshold  dB 
Maximum effort  
Fixed backoff probability  
Number of slots  slots 
Tunable Factors  Value 

Window length  slots 
Backoff factor  
Access Function  
Power control factor  
Power control factor  mW 