self - organizing behaviors of biological collectives have been subject to many scientific inquiries .they have also been enthusiastically applied to engineering problem solving as a new paradigm and methodology for decentralized , distributed problem solving by collaborative artificial agents .typical assumptions made in the existing literature are that the system components interact with each other mostly locally and make decisions at individual levels , which eventually leads to the emergence of non - trivial ( and potentially useful ) macroscopic behaviors .those models have been successful in reproducing various self - organizing patterns and adaptive functionalities .however , theoretical models used in earlier studies were predominantly focused on homogeneous physical collectives and animal populations . while their simplicity is by itself a virtue in some regard , they are often too simple to capture more complex phenomena seen in real - world biological collectives , such as multi - cellular organisms morphogenesis and physiology , termite colony building and maintenance , and growth and self - organization of human social systems .these systems operate with highly sophisticated within - system regulation mechanisms , or `` programs '' . what are common among those real - world complex biological collectives are _ heterogeneity of components _ , _ dynamic differentiation / re - differentiation of components _( i.e. , dynamic switching of component types / roles ) , and _ local information sharing among components _ that influence their differentiation .these _ morphogenetic principles _ are often absent in earlier models of biological collective behaviors , and they have not been fully utilized in engineering applications either . herewe define a _ morphogenetic collective system _ as a system made of a large number of components that self - organize to form nontrivial structures and behaviors using these morphogenetic principles .there is growing literature of analytical and numerical studies on self - organizing behavior of biological collectives .it is repeatedly reported that there are a small number of universal classes of collective behaviors , such as disordered , highly ordered , rotational , critical , and jamming patterns , as well as various forms of phase transitions between those classes .most of these results are based on the assumption that collectives are homogeneous in terms of their components kinetic and behavioral properties . within - population variationsare rarely considered in those theoretical models .there are some studies that considered the effects of heterogeneity within biological collectives .graves et al . studied mixed - species bird flocks in amazonia and created a computational model of them .more recently , couzin et al . studied self - sorting of a fish school caused by physical variations among individuals .his group also studied collective decisions of a swarm influenced by a small number of informed individuals , or leaders .we also proposed the swarm chemistry model to explore self - organization of swarms made of kinetically distinct types of particles .however , these studies still assumed that within - population variations are variations of fixed individual properties . none of them considered a more sophisticated form of dynamic , adaptive changes of behavioral rules of individuals within a population , potentially through local communication and information sharing .when one looks at real - world biological collectives , there are a number of examples where more complex forms of heterogeneous collectives produce highly intricate patterns and behaviors that look almost self - evidently `` designed '' by someone or something . such instances can be found at every scale in biology .for example , ben - jacob et al . reported very complex , heterogeneous , even intelligent , information processing and motion control taking place in bacteria societies .social insects are another well - studied example , where dynamic switching of different roles driven by local information sharing realizes highly efficient division of labor . among the most interesting and complex examplesare the incredibly sophisticated , large - scale mounds built and maintained by termites .a termite mound operates as if it were a carefully designed and fully integrated physiological system , inside which individual termites `` differentiate '' to play various different roles .the complexity of such real - world biological collectives were not fully captured in the earlier literature of collective behaviors mentioned above .as briefly reviewed above , the dynamics and capabilities of different types of morphogenetic collective systems are yet to be fully understood .recent increase of studies that incorporate at least part of the morphogenetic principles indicates the promising nature of this direction of research . in order to systematically study the effects of each of the morphogenetic principles , here we propose the following four distinct classes of morphogenetic collective systems , which are obtained by incrementally introducing morphogenetic principles to agents behavioral and communication capabilities ( fig .[ fourclasses ] ) :  1 .  _ homogeneous collectives _ , where agents behaviors are determined by a globally defined , uniformly applicable function of observations .2 .  _ heterogeneous collectives _ , where an agent s behavior is determined by a function of observations specified by the agent s static state or type .heterogeneous collectives with dynamic differentiation/ re - differentiation _ , where the heterogeneity of agent behaviors is created and dynamically maintained by transitions of agents internal states .agents state transitions are also determined by a function of observations and states ._ heterogeneous collectives with dynamic differentiation/ re - differentiation and local information sharing _ , where agents can share information ( internal states and their observations ) with local neighbors , in addition to all the above capabilities .our rationale in defining these four classes is that each morphogenetic principle requires the precedent one .namely , differentiation / re - differentiation requires multiple states or types of components ( heterogeneity ) , and information sharing would make sense only if agents could change their behaviors according to it ( differentiation / re - differentiation ) .we thus argue that these four classes should represent a natural , straightforward hierarchy of morphogenetic collective systems , arranged in the ascending order of their organizational complexity .    in this classification ,class a includes traditional collective behavior models , including boids and its variants that use homogeneous swarms .more recent models that use heterogeneous swarms belong to class b. examples that may be close to those of class c are the variants of swarm chemistry that implemented stochastic differentiation / re - differentiation of agents , though it was not driven by any state - transition function and thus not dynamical or adaptive. finally , real biological / social morphogenetic systems such as embryogenesis of multicellular organisms , colonies of eusocial insects and cities in human civilizations are most likely in class d. several models proposed in morphogenetic engineering also belong here . however , we are not aware of any well established model of swarm - based collective behaviors that belong to this class .to study qualitative and quantitative differences in possible morphologies and behaviors between the four classes , we have developed a mathematical model of morphogenetic collective systems by implementing new rule - based state transition and local information sharing capabilities in the swarm chemistry model .swarm chemistry is naturally suitable for this research task because it already can represent both homogeneous and heterogeneous collective systems in its model framework .  in the extended model ,the design of a swarm is specified in three parts : a recipe , a preference weight matrix , and a local information sharing coefficient .definitions of these parts are as follows :  recipe : : :  a list of different kinetic parameter settings for multiple swarm  states .each entry in a recipe is composed of a relative frequency of  a particular state within the swarm and its kinetic parameter settings  ( e.g. , local perception range , normal speed , strengths of kinetic  forces , etc . ) . this part of the specification is the same as in our  earlier studies .preference weight matrix : : :  a rectangular real - valued random matrix  where is the length of the recipe , i.e. , the number of  possible states of agents in a swarm .its contents represent how each  observation component ( defined later ) affects the agent s preference  for a particular state choice . local information sharing coefficient : : :  a real number in $ ] , which determines how much  information about the neighbors states and observations are to be  shared and incorporated into the agent s own state transition .each agent in a swarm in this extended model has its own state , in addition to position and velocity .the movement and state transition of agents is simulated as follows :  step 1 . : :  an agent computes its action ( acceleration ) based on its neighbors  relative positions and velocities using the standard swarm chemistry  simulation algorithm .: :  before making any actual movement , the agent also computes an  -dimensional _ observation vector _   that summarizes the situation the agent is in .more details of this  vector will be discussed later .: :  once all the agents computed their actions and observation vectors ,  each agent updates its velocity according to the computed  acceleration , and then moves using the updated velocity .: :  the agent computes a _ state preference vector _ , where   is the average of observation vectors  of other agents in the local neighborhood . if there are no other  agents found in the neighborhood , regardless  of .: :  the agent checks if the -th component of   , , is negative .if this is the  case , it means that its current state is not preferable in the current  situation , and therefore the agent attempts , with probability   , to choose a new state .a new state  will be chosen as the next value of , via a roulette  selection where is used as the selection  weight for state .the observation vector plays an important role in this model .while there are many choices for how to construct an observation vector , we used the following definition as an initial step of our investigation .the first components of are all 0 s except for the -th component that is set to 1 .this part embeds the information about the agent s current state into the vector so that it can be superposed and averaged with other agents states .the remaining five components of are as follows :  * : square of the relative distance from the average position of other agents in the neighborhood ( ) . is the perception range in the parameter set the agent is currently using .if there are no other agents nearby , this is set to 0 .* : square of the ratio of the agent s current velocity and the normal velocity ( ) in the parameter set the agent is currently using .* : square of the ratio of the neighbors average velocity ( ) and the normal velocity in the parameter set the agent is currently using . if there are no other agents nearby , this is set to 0 .* : square of the relative difference in velocity between the agent and its neighbors .if there are no other agents nearby , this is set to 0 .* : a constant term .although it is a highly constrained , mathematically stylized formulation , this extended swarm chemistry model can still describe a wide variety of morphogenetic collective systems , including the four classes proposed in this paper . specifically , including only one parameter set in and letting and will make a class a swarm . including multiple parameter sets in with and make a class b swarm .a class c swarm will be obtained by additionally adopting a non - zero matrix for while .finally , a class d swarm will be obtained by adopting non - zero and .we conducted systematic monte carlo simulations to see if there were any significant differences in the dynamics of morphogenetic collective systems among the four classes .specifications of swarm designs were configured as follows :  * the number of agents was fixed to 300 for all cases . *the number of possible agent states was set to 1 for class a swarms , or otherwise where is a random integer sampled from a poisson distribution with mean 1 .* recipe was created by sampling each kinetic parameter s value from a uniform distribution between 0 and the parameter s maximum value defined in .the relative frequencies of different parameter settings were determined by randomly dividing 300 into portions . *preference weight matrix was set to all 0 s for class a and b swarms . for class c and d swarms ,each component of was sampled from a normal distribution with mean 0 and standard deviation 0.1 .* local information sharing coefficient was set to 0 for class a , b and c swarms , while it was sampled from a uniform distribution between 0 and 1 for class d swarms .five hundred independent simulation runs were conducted for each class .each run was initialized with 300 agents whose positions were randomly distributed in a two - dimensional ( in arbitrary unit ) square area and whose velocities were set to 0 , and then simulated for 400 time steps . in each time step during the second half of the simulation ( ), the following properties were measured from the agents positions and velocities :  * average speed of the swarm as a whole ( , where denotes an average over all the agents hereafter unless noted otherwise ) . *average of absolute speed of agents ( ) . * average angular velocity of the swarm as a whole ( ) . *average distance of agents from center of mass ( ) . * average pairwise distance ( , where and are positions of two randomly sampled agents ; the average in this case was over 10,000 sampled pairs ) .in addition , topological properties of the swarm morphology were also measured . for this task, we constructed a network for each swarm at each time step during the measurement period by connecting agents that were spatially close to each other .more specifically , we first measured a characteristic `` neighbor '' distance for each agent by calculating the mean of its distances to its nearest neighbors .the agent then recognized all other agents within distance as its `` neighbors '' .once this was done for all agents , the pairs of agents that mutually recognized each other as `` neighbors '' were actually connected .we used and , which were empirically chosen to approximate cluster structures recognized visually by human experimenters .figure [ cluster - graph ] shows examples of this network construction process .( see text ) .c : resulting network where pairs of agents that mutually recognize each other as neighbors are connected .bottom : an example of network construction from a simulated swarm .d : original swarm snapshot .e : network constructed from agent positions in d. ]  for each swarm converted to a network , we measured the following topological properties :  * number of connected components . *average size of connected components .* homogeneity of sizes of connected components .this was measured by the normalized entropy in the distribution of sizes of connected components . if there was only one connected component , this was set to 1 .* size of the largest connected component . *average size of connected components smaller than the largest one .if there was only one connected component , this was set to 0 . * average clustering coefficient . * link density .altogether , we obtained time series of measurements from each simulation run .these time series were further summarized by calculating their respective mean and standard deviation over time . as a result ,the structure and behavior of each swarm was characterized by outcome variables .finally , a visual image of the swarm at the end of the simulation ( ) was also recorded for visual inspection of the swarm s topology .the simulator was implemented in python using networkx and pycx .the code is available upon request .simulation results are summarized in table [ resulttable ] , where medians of 24 outcome variables were shown for each class together with -values obtained using the kruskal - wallis median test between the four classes.sayama / swarmchemistry/. ] statistically significant differences were detected for most of the outcome variables , especially for topological outcome variables , which demonstrates the effectiveness of our network - based topology characterization method .most of the variables with statistical significance showed clear differences between class a ( homogeneous ) and other ( heterogeneous ) swarms .the temporal mean of the average clustering coefficient was the only outcome variable that did not show statistically significant differences between the classes .this metric may be primarily determined by constraints built in our simulation or network construction algorithms .+  [ resulttable ]  we found several notable patterns in table [ resulttable ] .first , the medians of class a swarms consistently took the lowest values among the four classes with regard to the temporal standard deviations of measurements ( lower half of table [ resulttable ] ) .this indicates that the interactions between different types of agents helped produce dynamic behaviors , causing temporal fluctuations of their macroscopic properties .second , there were clear differences between class b ( with no state transitions ) and classes c & d ( with state transitions ) regarding temporal standard deviations of the average distance of agents from the center of mass and the average pairwise distance .this is likely due to the fact that , while class b swarms tend to disperse into smaller clusters easily , agents of class c & d swarms can stay together and maintain spatially adjacent , coherent organization more often , because adaptive state transitions help initially incompatible agents assimilate into kinetically compatible types .finally , we noticed a consistent trend in a number of the outcome variables that the properties of class c & d swarms sat somewhere in between those in class a and class b. this could also be understood in that dynamic state transition , possibly driven by local information sharing , has enabled swarms to adaptively achieve coherence in their structures and behaviors .however , the results also indicate that the swarms in classes c & d did not simply turn into a more homogeneous state like class a ones , because nearly all the topological outcome variables showed significant differences between class a and classes c & d. therefore , this apparent trend found in morphogenetic collective systems in classes c & d must be understood not as simple homogenization but as an emergent shift of higher - level system properties .in this paper , we proposed a classification of morphogenetic collective systems based on the absence or presence of three morphogenetic principles : heterogeneity of components , dynamic differentiation / re - differentiation of components , and local information sharing among components .monte carlo simulations with an extended morphogenetic swarm chemistry model demonstrated that , while heterogeneity of components had a strong impact by itself on the structure and behavior of the swarms , the other two morphogenetic principles , i.e. , dynamic differentiation / re - differentiation of components and local information sharing , greatly contributed to the maintenance of spatially adjacent , coherent organization of swarms .interestingly , many outcome measurements of class c and d swarms fell somewhere in between those of class a and class b. this result indicates that the dynamic , adaptive state transition of components possibly driven by their mutual information sharing is playing an essential role in achieving structural and functional integration of biological and social collectives .the present study has several fundamental limitations .first , the dynamics of swarms were explored only through a limited number of random parameter sampling and analyzed only using simple median comparisons .it was a reasonable first step of exploration when nothing was known about the model , but this approach would not be able to characterize behavioral diversity and richness of each class of systems or to discover non - trivial behaviors that would be statistically rare but unique and interesting . to fully explore and understand the limit of dynamical diversity of each class , more sophisticated evolutionary or other population - based search methods should be conducted . now that we have 24 outcome metrics defined , we can try evolving morphogenetic collective systems toward a certain area in this metric space , to examine how closely the swarms of each class can achieve the target properties .second , the model we used in this study ( morphogenetic swarm chemistry ) was developed using somewhat arbitrary design decisions .the behavioral rules were exactly the same as those used in previous swarm models , which were limited to agents acceleration only .the state transition functions were defined in a rather simple and linear fashion using a product of a preferential weight matrix and an observation vector . moreover ,the variables included in the observation vector were chosen without substantial justification .we will need to consider adopting more open - ended , nonlinear forms of representations for observations , state transitions and agent behaviors .we plan to apply genetic programming or other symbolic evolutionary search methods to overcome this limitation in the future .finally , the scope of the experiments was limited only to `` free '' self - organization of collectives without any stimuli or constraints .exposing swarms to such external conditions and measuring their adaptive responses will further clarify the functional differences between different classes of morphogenetic collective systems .this material is based upon work supported by the us national science foundation under grant no .1319152 .bonabeau , e. , sobkowski , a. , theraulaz , g. , and deneubourg , j .- l .adaptive task allocation inspired by a model of division of labor in social insects . in lundh , d. , olsson , b. and narayanan a. , eds ., _ bio computation and emergent computing _ , pages 3645 .chuang , y. l. , dorsogna , m. r. , marthaler , d. , bertozzi , a. l. , and chayes , l. s. ( 2007 ) .state transitions and the continuum limit for a 2d interacting , self - propelled particle system ._ physica d _, 232 : 3347 .paley , d. a. , leonard , n. e. , sepulchre , r. j. , and couzin , i. d. ( 2008 ) .spatial models of bistability in biological collectives . in _ proceedings of the 46th ieee conference on decision and control _ , pages 48514856 .sayama , h. and wong , c. ( 2011 ) .quantifying evolutionary dynamics of swarm chemistry . in _ advances in artificial life ,ecal 2011 : proceedings of the eleventh european conference on artificial life _ , pages 729730. mit press .szabo , b. , szollosi , g. j. , gonci , b. , juranyi , zs ., selmeczi , d. , and vicsek , t. ( 2006 ) .phase transition in the collective migration of tissue cells : experiment and model ._ physical review e _ , 74 : 061908

Abstract: we studied the roles of _ morphogenetic principles_heterogeneity of components , dynamic differentiation / re - differentiation of components , and local information sharing among components in the self - organization of morphogenetic collective systems . by incrementally introducing these principles to collectives , we defined four distinct classes of morphogenetic collective systems . monte carlo simulations were conducted using an extended version of the swarm chemistry model that was equipped with dynamic differentiation / re - differentiation and local information sharing capabilities . self - organization of swarms was characterized by several kinetic and topological measurements , the latter of which were facilitated by a newly developed network - based method . results of simulations revealed that , while heterogeneity of components had a strong impact on the structure and behavior of the swarms , dynamic differentiation / re - differentiation of components and local information sharing helped the swarms maintain spatially adjacent , coherent organization .


since wireless channels are time varying , by properly choosing scheduling algorithm , it is possible to achieve multi - user diversity , which enhances the performance of a system considerably .most studies of scheduling algorithms have focused on optimizing the long term average throughput of users , i.e. , stability . from the stability point of view , it is important to design an algorithm which schedules the transmissions in such a way that the queues are stabilized at given offered loads .throughput optimal algorithms are considered as stable .definition of a throughput optimal algorithm is as follows : at any offered load if any other algorithm can stabilize a system , the designated algorithm can stabilize the system as well .for example ,maxweight ( mw ) and exponential ( exp ) algorithms are throughput optimal since they ensure stochastic stability of queues as long as such is feasible .  in other way, stability implies that average packet delay of users can not reach infinity .although stability is an important metric for designing a throughput optimal algorithm , for many delay - sensitive and guaranteed - rate traffic , it is not sufficient . for real - time applications such as voice and video ,we often need to ensure a stronger condition that the packet delay should be upper bounded or the long - term throughput should be lower bounded by some threshold with high probability .one approach to quantify the requirements of these delay - sensitive and guaranteed - throughput applications is to enforce constraints on the probability of the queue overflow . in other words ,we need to guarantee the smallest value of the probability that the largest queue length exceeds a given threshold , i.e. ,  ] .let  ] .if denotes the backlog of user at time , let  ] such that , } ] and = \mbox{pr}\left[q_i(t ) \ge \lambda_i - \lambda_i\zeta_i\right] ]. furthermore , denote by the total number of time slots when the channel is in state over the interval  ] is    the other functions of the system over the same interval are given by   , i\in[1,\cdots , n]\right).\end{aligned}\ ] ]  the state of the system is uniquely determined by the initial state .functions and drive the system , and , are determined by the scheduling algorithm and vice versa .the relationships of these functions over the time interval  ] to .since the arrival process of users follow poisson distribution , it is well known that the scaled arrival process of user at a particular time instant , has a sample path ldp with the following rate function  ,\ ] ]  where is a convex lower semi - continuous function on , and has the following properties    since the arrival processes of users are independent of each other , there exists a single decay rate function combining all users in the large deviation setting .this is because the decay rate of independent processes are additive .hence , the combined rate function of all users at a particular time instant can be written as    on the other hand , since the scaled channel state process follows i.i.d . in the time domain ,the rate function of this process can be given by    where , and \in\textit{p}_m ] with the rate function  ,~\mbox{if } ( \textbf{f}^b ( . ) \textbf{g}^b(.))\in ac , \\ \infty~\mbox{otherwise}\end{cases},\ ] ]  where denotes the set of absolute continuous functions in .the intuitive interpretation of this function is that for any set of trajectories in , the following relation holds :   \\\nonumber\ge \displaystyle\limsup_{b{\to}\infty}\frac{1}{b}\mbox{log}\textbf{p}\left[(\textbf{f}^b ( . ) , \textbf{g}^b(.))\in\lceil\right ] \\ \nonumber\ge -\inf_{(\textbf{f}^b ( . ) , \textbf{g}^b(.))\in\bar{\lceil}}j_t(\textbf{f}^b ( . ) , \textbf{g}^b(.)),\end{aligned}\ ] ]  here and are the interior and closure of set , respectively . for , we can have different sequences of , , and .for any value of , it is easy to verify that , , and are lipschitz - continuous , and their derivatives exist . as , there must exist a sequence of , , and that converge uniformly over the interval  ] indicates the long term growth rate of the backlog of user . if , is the minimum rate of growth of the backlog of the largest queue .now , define    * proposition 1 : * the stationary backlog process achieved by any scheduling algorithm satisfies the following relation    _ proof : _ from the definition of , it implies that provides a lower bound on the backlog of the largest queue , i.e. ,    for any , we can find such that .for any , we know that and .find such that }||(\textbf{f}(t ) , \textbf{g}(t))-(\textbf{f}_{\delta}(t ) , \textbf{g}_{\delta}(t))|| < \epsilon ] is a continuous function with respect to for any , we can write    where is a small number and . now, if we define , from ( [ eq : eqn3 ] ) , this in turn implies that    therefore    by the ldp definition of , we have   \\ \nonumber & \ge -\displaystyle\inf_{(\textbf{f } ( . ) , \textbf{g } ( . ) ) \in { \psi}_t({\textbf{f}}_{\delta } , { \textbf{g}}_{\delta } , \epsilon)}{\int}_0^t\left[l_{(\textbf{f})}(\dot{\textbf{f}}(t ) ) + l_{(\textbf{g})}(\dot{\textbf{g}}(t))\right]dt \\ \nonumber & \ge -{\displaystyle\int}_0^t\left[l_{(\textbf{f})}(\dot{\textbf{f}}_{\delta}(t ) ) + l_{(\textbf{g})}(\dot{\textbf{g}}_{\delta}(t))\right]dt \\ \nonumber & = -t\left[l_{(\textbf{f})}({\textbf{y}}_{\delta } ) + l_{(\textbf{g})}({\bm{\gamma}}_{\delta})\right ] \\\nonumber & = -\left(1 + \epsilon{\epsilon}_1\right)(i_{opt } + \delta).\end{aligned}\ ] ]  since , and are arbitrarily small , we can conclude the proof of _ proposition 1_.   the large deviation philosophy implies that rare events occur in the most likely way .hence , the probability of the queue overflow is determined by the smallest cost among all sample paths that overflow .so , the decay rate of the queue overflow probability is the minimum cost among all fluid sample paths that overflow . to prove the lower bound of this decay rate , we consider that the system starts at time 0 , instead of considering the entities at the stationary system .consider a certain time .let denote the set of fluid sample paths on the interval  ] into ( integer number ) subintervals such that the length of each subinterval is .hence , ] .then , we divide the simplex of all vectors representing the probability distributions on the set of channel states into ( integer number ) non - intersecting subsets ( `` bins '' ) such that the oscillation ( difference between the maximum and minimum ) of within the closure of each bin is at most . if necessary, we increase / decrease the value of so that this condition is satisfied uniformly over all bins .  for any , and for the time interval ] .we know that the derivative of ( ) resides in each of the bins ( described above ) , and the derivative of ( ) is in each of the bins . consequently , the derivatives of the function over the time interval  ] possible combinations . for any ,consider one fixed aggregate bin , of which  ] , we set .then , for and for any , is the set of arrival and channel rate processes such that the corresponding backlog process satisfies .obviously , each component of is the instance of vector function .consequently , we have   \\ \le \displaystyle\limsup_{b \to \infty}\frac{1}{b}\mbox{log}\mbox{pr}\left[(\textbf{f}^b ( . ) , \textbf{g}^b ( . ) ) \in \cup_{b=1}^{\infty}{\lceil}^b\right]\end{aligned}\ ] ]  by the ldp definition for ( ) in ( [ eq : eqn7 ] ) , we have   \\ \le - \inf_{(\textbf{f } ( . ) , \textbf{g } ( . ) ) \in \overline{\cup_{b=1}^{\infty}{\lceil}^b}}\displaystyle\int_0^t\left[l_{(\textbf{f})}(\dot{\textbf{f}}(t ) ) + l_{(\textbf{g})}(\dot{\textbf{g}}(t))\right]dt + 2{\epsilon}_0t.\end{aligned}\ ] ]  since , it turns out   \\ \le - \inf_{(\textbf{f } ( . ) , \textbf{g } ( . ) ) \in \overline{\cup_{b=1}^{\infty}{\lceil}^b}}\displaystyle\int_0^t\left[l_{(\textbf{f})}(\dot{\textbf{f}}(t ) ) + l_{(\textbf{g})}(\dot{\textbf{g}}(t))\right]dt + 2{\epsilon}_0t.\end{aligned}\ ] ]  to prove _ proposition 2 _ , it is sufficient to show that the right hand side of ( [ eq : eqn6 ] ) is no greater than that of ( [ eq : eqn8 ] ) . for each , we can find such that   - 2{\epsilon}_nt \\  & < \displaystyle\inf_{(\textbf{f } ( . ) , \textbf{g } ( . ) ) \in \overline{\cup_{b=1}^{\infty}{\lceil}^b}}\displaystyle\int_0^t\left[l_{(\textbf{f})}(\dot{\textbf{f}}(t ) ) + l_{(\textbf{g})}(\dot{\textbf{g}}(t))\right]dt - 2{\epsilon}_0 t + \frac{1}{n}.\end{aligned}\ ] ]  since is equicontinuous , we can find a subsequence that has a uniform derivative on the time - interval  ] is satisfied .  for each ,let be the corresponding backlog process for .similar to , is equi - continuous and has a uniform non - negative derivative on the interval  ] achieved by our scheduling algorithm . in this context, we would like to study the relationship of the system components and their derivatives in granular time - interval . to understand the behavior of the system ,scaled processes are already derived and given by ( [ sec : scaled - procs ] ) . to interconnect the outcome of scheduling policy with these scaled processes , we study the scheduling rule at time instant for any and . after re - arranging the scheduling rule for user at time instant , we have }e^{-\frac{q_i(bt)}{q^{th}}} ] .since the scaled queue length part of our scheduling rule is proportional to some real number multiple of the original non - scaled queue length , it is sufficient to find the relationship between the system components and the scheduling policy at the scaled time instant instead of further refinement ._ lemma 1 _ summarizes these relationships followed by the corresponding proof . in a non - overloaded state of the system, we provide the structure of the sample path to overflow event over a finite time interval in _ theorem 1 _ , and then in _ theorem 2 _ , we prove that the value of of our scheduling rule affects the decay rate of the largest queue overflow probability .  * lemma 1 : * at the scaled time instant  ] .since all functions are lipschitz - continuous , their derivatives in ( [ eq : derivs ] ) exist .the relations in ( [ eq : rel1 ] ) , ( [ eq : rel3 ] ) and ( [ eq : rel4 ] ) are a simple consequence of the relations in ( [ eq : orig - rel1 ] ) and ( [ eq : orig - rel2 ] ) .since our objective is to study the path to the largest queue overflow event , we denote the largest queue at scaled time instant by . since at scaled time instant , exists , the relation in ( [ eq : rel2 ] )is satisfied consequently .consider a scaled time - interval  ] . during this unscaled time - interval , at any time instant , if the channel state is selected , user is chosen for service based on the rule }e^{-\frac{q_i^b(t)}{q^{th}}} ] for a non - overloaded system    _ proof : _ let be the channel - user assignment matrix for the best possible non - overloaded system , and be the corresponding service rate vector such that . at time instant , consider a subset of users , the users of which satisfy .let us assume that channel state is sampled at time instant , such that and . then , using an argument similar to that in  ( at the end of section ) , we can establish the following fact    multiplying on both sides for user , we obtain    now , implies that is close to , and is close to .furthermore assume that stochastic matrix is the resultant outcome for the actual arrival rate distribution of the system .therefore , for all sufficiently small values ( i.e. , ) of , we have    after multiplying for channel state , and then summing the relation in ( [ eq : eqn11 ] ) over all channel states , we obtain    sufficiently small values ( i.e. , ) of further implies   \le \sum_{i{\in}\textbf{n}^*}[\lambda_i - v_i^ * ]  < 0.\ ] ]  at time instant  ] at a non - overloaded state of the system    _ proof : _ the proof is as same as _ lemma 9.3 _ of  .* theorem 1 : * in a non - overloaded state of the system , for any , and , let us denote  \},\ ] ]  the aforementioned definition implies that  }q_*(t ) \ge \delta_2\right ) \le -k(\delta_1 , \delta_2 , t).\ ] ]  based on the above definitions , given some fixed constant , the value of grows linearly with .more precisely , for any , there exists such that for a moderately large value of and , .  _proof : _ the proof of this theorem can be proved by the statement of _ lemma 3_.  * theorem 2 : * at any time instant , suppose and are the arrival rate and channel state vectors . ] .consequently , be the set of all possible service rate vectors for all users .furthermore , we assume that . if is the growth rate of the largest queue , and is the service rate vector , the value of resides in the solution of the auxiliary optimization problem in ( [ eq : opt1 ] ) . \\ & \mbox{subject to } \sum_{i}e^{-q_i / q^{th } } \le a,~\mbox{is a fixed constant and } > 0 .\end{aligned}\ ] ]  the optimization problem in ( [ eq : opt1 ] ) has the following properties  \(i ) it has the following structure , which is equivalent to the rule of our scheduling algorithm .    \(ii ) the resultant value of is affected by the value of , and the optimal decay rate achieved by our scheduling algorithm can be obtained by tweaking this variable .\(iii ) it is equivalent to the optimization problem in ( [ eq : opt2 ] ) . and , the value of the problem in ( [ eq : opt1 ] ) is equal to multiple of the value in ( [ eq : opt2 ] ) . \\ & \mbox{subject to } v_i \ge 0 , \forall{i}.\end{aligned}\ ] ]  _ proof : _ if we redefine the variables , the optimization problem in ( [ eq : opt1 ] ) can be written as   \\ \nonumber & \mbox{subject to } \sum_iz_i \le a \\ & z_i \ge 0 , \forall{i}.\end{aligned}\ ] ]  note the following property of the problem in ( [ eq : opt3 ] )    the aforementioned function in ( [ eq : func1 ] ) is convex .the inner problem ( the min part ) in ( [ eq : opt3 ] ) is then the concave function , and then we can write the convex problem in ( [ eq : opt4 ] ) as    subject to the constraints in ( [ eq : opt3 ] ) . taking the lagrangian of the problem in ( [ eq : opt4 ] ) over * z* , we have    where and , are the lagrange multipliers . for any optimal solution of the problem , there exist some fixed and for which the following conditions are satisfied    where    this is because a vector is a subgradient of at point if and only if . due to the duality property , must be positive that implies are positive , and since is positive .since , we can say that the resultant is the real number multiple of .consequently , from the definition , can be renamed as the service rate vector .furthermore , from the definition of the growth rate of the largest queue , .this proves the statement of ( i ) in _ theorem 2_. consequently , at the unique optimal solution , the value of the problem in ( [ eq : opt3 ] ) is   = \displaystyle\sum_iz_i^*\left[\lambda_i - b_i\right ] = a\pi.\ ] ]  now , we want to see how the value of affects the value of that eventually affects the value of .intuitively , the larger the value of , the resultant is such that it gives more priority to the value of instead of . on the other hand , the smaller the value of , the resultant is such that it gives more priority to the value of * z*.consequently , the values of and are affected as well . to summarize , consider and are two possible values of , where .for and , if the resultant values of are and , respectively , we can say that .the optimal value of ( the minimum one ) resides in choosing the proper value of . generally , the smaller the value of ( while ) , the better the value of .this concludes the statement of ( ii ) in _ theorem 2_.  the proof of the statement in ( iii ) is as follows .we know that \le \max_i[\lambda_i - e^{\{-\frac{v_i}{\max_iv_i } + \rho_1 + \rho_2\ } } ] , \forall{i} ] . for this rule ,similar to the above two cases , we plot the decay rate and buffer overflow probability in fig . [fig : mw - dr ] and fig .[ fig : mw - of - prob ] , respectively .table v(a ) and table v(b ) present the user - channel state distribution matrix for and , respectively . unlike the exp rule , with this rule , larger value of makes the scheduling algorithm to be more dependent on the instantaneous queue length comparing with the instantaneous channel rate .however , as we mentioned in the previous paragraphs , the scheduling algorithm that is solely dependent on queue length is not throughput optimal , can not provide optimal decay rate . to achieve the optimality, one needs to choose the proper value of that provides the proper balance between the instantaneous queue length and the instantaneous channel rate dependence of the scheduling algorithm .consequently , the optimal decay rate is achieved at .table v(a ) and table v(b ) further emphasizes this observation . in fig .[ fig : mw - of - prob ] , it is further evident that the better the decay rate , the better the buffer overflow probability .the above simulation results discussed raise some issues on the applicability of the large deviation principle as the probability of any event is an exponential function of its decay rate . from our analytical study and also from  , buffer size that is close to fits the characteristics of ldp .this study does not provide much information on what buffer level is enough for the asymptotic system behavior to be accurate .furthermore , the ldp study only specifies the decay rate of the probability of any event .if the factor in front of the exponential term is unknown , the exact value of the probability from the ldp study is not possible .hence , one needs to be careful while comparing the performance predicted by a ldp with the actual performance ( e.g. , buffer overflow probability ) of any scheduling algorithm .  to better understand the results, we plot the state space of the simulation ( our scheduler ) in fig .[ fig : our - dec - reg - qth-2 ] and fig .[ fig : our - dec - reg - qth-10 ] for and , respectively .we project the length of two chosen queues and on the x - axis and y - axis , respectively . based on the scheduling decision ,the state space is divided into decision regions . in region , user is scheduled based on its queue length irrespective of the channel state . in the similar manner , no matter the channel state , in region , is served . in region , either or is served based on their channel state and queue length .boundary of the decision regions is determined by the scheduling policy .the tiny circles in the figures are the states that have been visited by the system during the simulation .  .]  ]  region and region are referred to as max - queue regions as in these regions , scheduling decisions are only made based on the queue length . whereas , region is named as the max - rate region as the user instantaneous rate is considered while making the scheduling decision in this region . for our scheduling algorithm , as we decrease , the boundaries between the decision regions tend to move to the diagonal line .this trend has two implications .first , as the decision boundaries approach the diagonal line , the algorithm places more emphasis on reducing the largest queue .apparently , with this setup , the decay rate of the queue overflow probability should be improved significantly .however , the second effect of decreasing is the reduction of the max - rate region . as a result , for smaller value of queue length , it is less likely that the system falls into the max - rate region . with smaller value of ,the algorithm is unlikely to take the advantage of the increased capacity at small queue lengths which leads to the tendency for the queues to grow .the area of the max - rate region introduces the tradeoff in increasing / decreasing the largest queue length . for this reason , as observed in fig .[ fig : our - dr ] , the decay rate of the overflow probability for is smaller than that with .  . ]  similar to the state space plots achieved from the simulation of our scheduling algorithm , we show the decision boundaries of the state space derived from the simulation of the exp scheduler in fig .[ fig : exp - dec - region ] for different values of .the slight variation of the decay rate for different values of can be understood when we see different boundary lines . from this figure and the plot in fig .[ fig : our - dr ] , this is evident that with larger values of , the larger max - rate region may not be useful in reducing the largest queue length .  .]  for different values of , we also plot the decision boundaries of the simulated state space obtained by the mw scheduling algorithm .we argued previously that the mw rule provides more emphasis on the largest queue length for larger values of , and fails to take the advantage of the instantaneous channel rate .this observation is further proved in this figure . however , with smaller values of , the algorithm exaggeratedly gives more emphasis on the instantaneous use rate , and hence the largest queue length increases consequently . as a result , the best decay rate of the overflow probability is obtained at .in this paper , we have analyzed the performance of the scheduling algorithm proposed in  . because of the complex coupling between input and output metrics of a scheduling algorithm , it is difficult to analyze such algorithm using conventional methods .since the scheduling algorithm is specifically designed for qos based traffic , we have mapped the probability that the scheduling algorithm does not meet the qos bound to the queue overflow probability .then , we have used ldp to determine the bound of this probability . we have proved that for certain value of the parameter , the scheduling algorithm converges to the optimal possible algorithm which can reduce the queue overflow probability to the smallest possible level . finally , through the numerical results, we have proved our analytical study comparing with other scheduling algorithms .r. ruby , v. c. m. leung , and d. g. michelson , `` uplink scheduler for sc - fdma - based heterogeneous traffic networks with qos assurance and guaranteed resource utilization , '' _ ieee trans . veh ._ , vol . 64 , no . 10 , pp . 47804796 , oct 2015 .      a. l. stolyar , `` large deviations of queues sharing a randomly time - varying server , '' _ queueing syst ._ , vol .59 , no . 1 ,135 , may 2008 .[ online ] .available : http://dx.doi.org/10.1007/s11134-008-9072-y    m. andrews , k. kumaran , k. ramanan , a. stolyar , r. vijayakumar , and p. whiting , `` scheduling in a queuing system with asynchronously varying service rates , '' _ probab ._ , vol . 18 , no . 2 , pp .191217 , apr 2004 .[ online ] .available : http://dx.doi.org/10.1017/s0269964804182041        s. meyn , `` stability and asymptotic optimality of generalized maxweight policies , '' _siam j. control optim ._ , vol .47 , no . 6 , pp .32593294 , 2009 .[ online ] .available : http://link.aip.org/link/?sjc/47/3259/1

Abstract: in this paper , we study the stability of light traffic achieved by a scheduling algorithm which is suitable for heterogeneous traffic networks . since analyzing a scheduling algorithm is intractable using the conventional mathematical tool , our goal is to minimize the largest queue - overflow probability achieved by the algorithm . in the large deviation setting , this problem is equivalent to maximizing the asymptotic decay rate of the largest queue - overflow probability . we first derive an upper bound on the decay rate of the queueoverflow probability as the queue overflow threshold approaches infinity . then , we study several structural properties of the minimum - cost - path to overflow of the queue with the largest length , which is basically equivalent to the decay rate of the largest queue - overflow probability . given these properties , we prove that the queue with the largest length follows a sample path with linear increment . for certain parameter value , the scheduling algorithm is asymptotically optimal in reducing the largest queue length . through numerical results , we have shown the large deviation properties of the queue length typically used in practice while varying one parameter of the algorithm .   submitted paper  scheduling , heterogeneous traffic , large deviation principle .


we consider a channel , termed an _ individual channel _ , where no specific probabilistic or mathematical relation between the input and the output is assumed .this channel is an extreme case of an unknown channel .achievable rates are characterized using only the input and output sequences , which capture the actual ( a - posteriori ) channel behavior .this point of view is similar to the approach used in universal source coding of individual sequences where the goal is to asymptotically attain for each sequence the same coding rate achieved by the best encoder from a model class , tuned to the sequence .this framework is an evolution of those considered in shayevitz and feder and eswaran et . and is presented in more detail in our papers and , together with the relevant background .we will give a brief introduction below .the setting we consider includes a single encoder receiving a message to transmit and emitting symbols and a decoder receiving a sequence of symbols and attempting to reconstruct the message . in the present paperthe input and output symbols are real - valued vectors , i.e. and .the relation between ^t ] is unknown to the encoder and decoder .we consider two communication scenarios : with feedback ( possibly imperfect ) and without feedback .for the case in which there is no feedback the communication system transmits in a constant rate , and outage is unavoidable , i.e. one can not guarantee a small probability of error in all circumstances. in the case feedback exists , the communication rate may be dynamically adapted and outage may be prevented . in both caseswe assume common randomness exists in the encoder and the decoder .the results in the current paper extend the previous results , yet only for the first case , of transmission in a constant rate .the performance is measured by a rate function representing an empirical measure of the achievable rate between the channel input and channel output , over channel uses . in examples here and in , can be viewed as the mutual information achieved in a certain family of statistical models ( in the current scope , all zero mean gaussian channels ) , when the model parameters match the empirical ones . in communication without feedbackwe say that a given rate function is achievable with an input distribution if for large block size , it is possible to communicate at any rate and an arbitrarily small error probability is obtained whenever . the communication system is required to emit blocks with probability distribution , which is possible due to the use of randomization . by placing this additional constraintwe leave aside the question of adapting the input distribution , so that the current framework attempts at achieving the empirical `` mutual information '' rather than the empirical `` capacity '' .another reason for the fixed prior is avoiding degenerate systems which may transmit only `` bad '' sequences with low ( or zero ) .this constraint is further discussed in , section viii.c .the main result of this paper is that for the multiple - input multiple - output ( mimo ) channel ( i.e. with transmit and receive antennas ) the rate function defined below is asymptotically achievable , in the fixed rate case : where the matrix denotes the channel input over symbols , and the matrix denotes the output . , and ^t [ \mt x \mt y] ] then from the same considerations we will have , therefore the factors cancel out and 3 .[ property3 ] * symmetry : * [ theorem : continuous_nonadaptive_mimo1 ] given the channel , define the input over symbols as an matrix , and the output as an matrix .let , and ^t [ \mt x \mt y] ] .we perform a qr decomposition of and in order to obtain more friendly expressions . as a reminder, qr decomposition of a matrix ( with and upper triangular ) is performed by gram - schmidt process .we start from the left column of and work our way to the last one . at each timewe take a column of and split it to the part which can be represented by a linear combination of the columns to the left of it ( equivalently , to the columns of already generated ) , and the `` innovation '' , i.e. the part which is orthogonal to the subspace generated by the previous columns . the vector representingthe innovation is normalized , and becomes the respective column of , and its power becomes the diagonal element in .the coefficients representing the part of the vector which is in the subspace of previous columns become the elements of above the diagonal .another important property of qr decomposition is that the determinant of can be written in terms of the diagonal elements in : .now define the diagonal of the upper triangular matrix in the qr decomposition of the matrices , and respectively to be the vectors , and  ] .the lengths of the vectors are respectively , so that they overlap with the columns of and in the matrix .we have : note that the factors cancel out because the matrix dimensions are and in the nominator and in the denominator . since the gram - schmidt process operates sequentially from the first column to the last , and the first columns of and are equal , we will have .therefore we can write : that and both relate to the same vector , the -th column of .the ratio is the ratio between the innovation of the -th column of with respect to the subspace spanned by previous columns of alone ( nominator ) or these columns together with the columns of ( denominator ) . obviously from this reason ( and therefore - property [ property1 ] ) .the key observation in this derivation is as follows : consider a sequential drawing of the columns of and calculation of the factors .since the -th column of is chosen isotropically and independently of the previous columns , the value of previous columns does not affect the distribution of the innovations ( only the number of dimensions in previous columns does ) . using this observation which we will prove subsequently, we would be able to break represented as the expected value of a product ( [ eq : r_as_ad ] ) into a product of expected values ( equations ( [ eq : l_prod_induction])-([eq : l_as_prod_d ] ) ) , and the proof is completed by a ( tedious ) calculation of these expected values .  to show the independence of in previously drawn values , denote by a matrix including the columns to of , and by the -th column .define a a unitary matrix whose first columns span the subspace spanned by the first columns of , its next columns extend this subspace to cover also the columns subspace of , and the next columns complete it to an orthonormal basis .this matrix does not depend on and specifically on the column .we assume that the columns of are linearly independent ( we will relax this assumption later ) . also , in probability one , assuming , the columns of are linearly independent of each other and of the columns of . to see this ,it is easy to show that the projection of each column in any direction orthogonal to the subspace already spanned by previous ones ( including ) , is also gaussian therefore has probability to be , as long as there exists such an orthogonal vector , i.e. the number of previously generated vectors is smaller than .now define . since also .the first elements of represent the projections of to the subspace spanned by previous columns of , and the next elements represent the projections to the subspace spanned by columns of .so collects the energy of all elements except the first , and collects the energy of all elements except the first . to see this formally , in the gram - schmidt process the coefficients of the projection of on the subspace spanned by are , andthe projection itself is , therefore the innovation is . since and we have  \vr v_i \right\|^2 = \left\| \left [ \begin{array}{c } 0 \\ { \mt q_{i}^n}^t \vr x_i \end{array } \right ]  \right\|^2 = \left\|  \vr z_{i}^n\right\|^2 $ ] and similarly , .therefore are independent of and the previous columns of , and can be given by norms over parts of a gaussian i.i.d .vector of length . defining where the equality is due to the independence shown above, we have for any : = \\= e \left [ \prod_{i=1}^{k-1 } \left ( \frac{a_i}{d_i } \right)^{\gamma n } \cdot e \left ( \left .\left ( \frac{a_k}{d_k } \right)^{\gamma n } \right| \mt x_{1}^{k-1 } \right ) \right ] = \\= e \left [ \prod_{i=1}^{k-1 } \left ( \frac{a_i}{d_i } \right)^{\gamma n } \cdot d_k \right ] = e \left ( \prod_{i=1}^{k-1 } \left ( \frac{a_i}{d_i } \right)^{\gamma n } \right ) \cdot d_k\end{gathered}\ ] ] therefore by induction : ^{\frac{\gamma \cdot n}{2 } } = e \prod_{i=1}^{t } \left ( \frac{a_i}{d_i } \right)^{\gamma n } \stackrel{(\ref{eq : l_prod_induction})}{= }\prod_{i=1}^{t } d_i\ ] ]  now we bound ( using the previously defined gaussian vector ) : where in ( a ) we used independent chi - squared distributed , and in ( b ) we changed variables from to , with inverse transformation and jacobian .the first integral in the expression above evaluates to : by definition of .the second integral behaves like near and like at . therefore it will exist ( converge ) iff the power of near is larger than and at is smaller than .the first condition is .the other condition always holds since . note that since the power of is larger by more than 1 than the power of it is positive ( when the first condition holds ) .therefore we can bound :    combining ( [ eq : d_i_derivation ] ) , ( [ eq : w_integral ] ) and ( [ eq : v_integral_bound ] ) we obtain :   since results in a rate loss of , and is superexponential in , we would like to express more explicitly the dependence on . using with , can obtain the bound therefore and   substituting the above into ( [ eq : chernoff_error_bound ] ) proves lemma [ lemma : pairwise_mimo ] for .the two assumptions on the parameters of the problem we have made in order for to be bounded are ( a ) which was needed in order that each new column of would not be spanned by the previous columns and the columns of in probability 1 , and ( b ) , is needed for the existence of .suppose now that .using the cholesky decomposition we can define a coloring matrix , so that and . since by property [ property2 ] the rate function is invariant to a linear transformation of we would have , therefore if lemma [ lemma : pairwise_mimo ] holds with respect to the white signal it also holds with respect to . with regard to the assumption that the columns of are linearly independent : if they are not , then the rate function is defined with respect to a smaller matrix containing only the independent columns . comparing with a full rank , the random variables increase ( i.e. ) due to the smaller dimension of , therefore and the lemma still holds* comparison with the siso case : * comparing lemma [ lemma : pairwise_mimo ] with lemma 4 of for the siso case , which is proven by a direct calculation , the bound here is slightly worse due to the limitation which stems from the use of the chernoff bound . * comparison with mimo capacity : * the scheme above achieves the mutual information of a gaussian mimo channel but not its capacity .achieving the capacity requires adaptation of the input distribution , which for the known awgn channel is performed by svd and water pouring .the strength of the scheme is in the lack of any assumptions about the probability distribution , which make it applicable for example for non gaussian noise or one that depends on the transmitted signal .  * exploiting temporal correlation : * in the current results , as in previous ones , the rate function depends on the zero order empirical probability , and lacks the ability to exploit temporal correlation .however the results can be used to exploit such correlation in the siso or mimo channel , in a crude way , by applying the scheme on blocks of channel uses . the rate function over blocks is always superior to the single letter case , and the penalty is an increase in the fixed redundancy .  * using empirical covariance instead of correlation : * when the matrices in ( [ eq : remp_mimo ] ) are replaced with the empirical correlation ( where ) , the derivation is similar , except projection on an additional dimension ( the all - ones vector ) precedes the other projections .the results are the same with a loss of one dimension : and are required , and there is a small variation in .  *the complex mimo channel : * the results easily extend to the _ complex_-valued mimo channel , using the same technique .the main difference is a double number of degrees of freedom in the derivation of , which doubles the rate compared to equation [ eq : remp_mimo ] .* adaptivity : * in we presented a communication scheme using a low rate feedback , which dynamically adapts the transmission rate and achieves the rate functions without outage .such schemes are of higher practical interest .it is possible to show that the adaptive scheme of achieves of ( [ eq : remp_mimo ] ) up asymptotically vanishing redundancy , and up to a set of sequences having vanishing probability .o. shayevitz and m. feder , `` achieving the empirical capacity using feedback : memoryless additive models'',ieee transactions on information theory , vol.55 no.3 , march 2009 , pp.1269 -1295 k. eswaran , a.d .sarwate , a. sahai , and m. gastpar , `` zero - rate feedback can achieve the empirical capacity , '' ieee transactions on information theory , vol.58 , no.1 , january 2010 y. lomnitz and m. feder , `` communication over individual channels , '' arxiv:0901.1473v1 [ cs.it ] , 11 jan 2009 , http://arxiv.org/abs/0901.1473v1 y. lomnitz and m. feder , `` feedback communication over individual channels , '' ieee international symposium on information theory ( isit ) , 2009 , pp.1506 - 1510 , june 28 2009-july 3 2009 i. telatar , `` capacity of multi - antenna gaussian channels , '' at&t technical memorandum , june 1995 a. goldsmith , s.a .jafar , n. jindal , and s. vishwanath , `` capacity limits of mimo channels , '' ieee journal on selected areas in communications , vol . 21 , no .5 , june 2003

Abstract: we consider the problem of communicating over a multiple - input multiple - output ( mimo ) real valued channel for which no mathematical model is specified , and achievable rates are given as a function of the channel input and output sequences known a - posteriori . this paper extends previous results regarding individual channels by presenting a rate function for the mimo individual channel , and showing its achievability in a fixed transmission rate communication scenario .