we consider wireless communication networks where nodes are able to harvest energy from nature .the nodes may harvest energy through solar cells , vibration absorption devices , water mills , thermoelectric generators , microbial fuel cells , etc . in this work ,we do not focus on how energy is harvested , instead , we focus on developing transmission methods that take into account the _ randomness _ both in the _ arrivals of the data packets _ as well as in the _ arrivals of the harvested energy_. as shown in fig . [ fig : system1 ] , the transmitter node has two queues .the data queue stores the data arrivals , while the energy queue stores the energy harvested from the environment . in general , the data arrivals and the harvested energy can be represented as two independent random processes .then , the optimal scheduling policy becomes that of adaptively changing the transmission rate and power according to the instantaneous data and energy queue lengths .while one ideally should study the case where both data packets and energy arrive randomly in time as two stochastic processes , and devise an _ on - line _ algorithm that updates the instantaneous transmission rate and power in _ real - time _ as functions of the current data and energy queue lengths , this , for now , is an intractable mathematical problem . instead , in order to have progress in this difficult problem , we consider an idealized version of the problem , where we assume that we know exactly when and in what amounts the data packets and energy will arrive , and develop an optimal _ off - line _ algorithm .we leave the development of the corresponding _ on - line _algorithm for future work .specifically , we consider a single node shown in fig .[ fig : bit_arri ] .we assume that packets arrive at times marked with and energy arrives ( is harvested ) at points in time marked with . in fig .[ fig : bit_arri ] , denotes the number of bits in the arriving data packet , and denotes the amount of energy in the energy arrival ( energy harvesting ) .our goal then is to develop methods of transmission to minimize the time , , by which all of the data packets are delivered to the destination .the most challenging aspect of our optimization problem is the _ causality _ constraints introduced by the packet and energy arrival times , i.e. , a packet may not be delivered before it has arrived and energy may not be used before it is harvested .the trade - off relationship between delay and energy has been well investigated in traditional battery powered ( unrechargeable ) systems .references investigate energy minimization problems with various deadline constraints .reference considers the problem of minimizing the energy in delivering all packets to the destination by a deadline .it develops a _ lazy scheduling algorithm _, where the transmission times of all packets are equalized as much as possible , subject to the deadline and causality constraints , i.e. , all packets must be delivered by the deadline and no packet may be transmitted before it has arrived .this algorithm also elongates the transmission time of each packet as much as possible , hence the name , _ lazy _ scheduling . under a similar system setting, proposes an interesting novel calculus approach to solve the energy minimization problem with individual deadlines for each packet .reference develops dynamic programming formulations and determines optimality conditions for a situation where channel gain varies stochastically over time .reference considers energy - efficient packet transmission with individual packet delay constraints over a fading channel , and develops a recursive algorithm to find an optimal off - line schedule .this optimal off - line scheduler equalizes the energy - rate derivative function as much as possible subject to the deadline and causality constraints .references and extend the single - user problem to multi - user scenarios . under a setting similar to ,we investigate the average delay minimization problem with a given amount of energy , and develop iterative algorithms and analytical solutions under various data arrival assumptions in .references investigate delay optimal resource allocation problems under various different settings .references consider average power constrained delay minimization problem for a single - user system , while minimize the average delay through rate allocation in a multiple access channel .  in this paper, we consider a single - user communication channel with an energy harvesting transmitter .we assume that an initial amount of energy is available at .as time progresses , certain amounts of energies will be harvested .while energy arrivals should be modeled as a random process , for the mathematical tractability of the problem , in this paper , we assume that the energy harvesting procedure can be precisely predicted , i.e. , that , at the beginning , we know exactly when and how much energy will be harvested . for the data arrivals , we consider two different scenarios . in the first scenario, we assume that packets have already arrived and are ready to be transmitted at the transmitter before the transmission starts .in the second scenario , we assume that packets arrive during the transmissions .however , as in the case of energy arrivals , we assume that we know exactly when and in what amounts data will arrive .subject to the energy and data arrival constraints , our purpose is to minimize the time by which all packets are delivered to the destination through controlling the transmission rate and power .this is similar to the energy minimization problem in , where the objective is to minimize the energy consumption with a given _ deadline _ constraint . in this paper , minimizing the transmission completion time is akin to minimizing the deadline in .however , the problems are different , because , we do not know the exact amount of energy to be used in the transmissions , even though we know the times and amounts of harvested energy .this is because , intuitively , using more energy reduces the transmission time , however , using more energy entails waiting for energy arrivals , which increases the total transmission time .therefore , minimizing the transmission completion time in the system requires a sophisticated utilization of the harvested energy .to that end , we develop an algorithm , which first obtains a good lower bound for the final total transmission duration at the beginning , and performs rate and power allocation based on this lower bound .the procedure works progressively until all of the transmission rates and powers are determined .we prove that the transmission policy obtained through this algorithm is globally optimum .we assume that there are a total of bits available at the transmitter at time .we also assume that there is amount of energy available at time , and at times , , , , we have energies harvested with amounts , , , , respectively .this system model is shown in fig .[ fig : system ] .our objective is to minimize the transmission completion time , .we assume that the transmitter can adaptively change its transmission power and rate according to the available energy and the remaining number of bits .we assume that the transmission rate and transmit power are related through a function , , i.e. , .we assume that satisfies the following properties : i ) and as , ii ) increases monotonically in , iii ) is strictly concave in , iv ) is continuously differentiable , and v ) decreases monotonically in .properties i)-iii ) guarantee that exists and is strictly convex .property v ) implies that for a fixed amount of energy , the number of bits that can be transmitted increases as the transmission duration increases . it can be verified that these properties are satisfied in many systems with realistic encoding / decoding schemes , such as optimal random coding in single - user additive white gaussian noise channel , where .assuming the transmitter changes its transmission power times before it finishes the transmission , let us denote the sequence of transmission powers as , , , , and the corresponding transmission durations of each rate as , , , , respectively ; see fig .[ fig : system2 ] .then , the energy consumed up to time , denoted as , and the total number of bits departed up to time , denoted as , can be related through the function as follows : where .then , the transmission completion time minimization problem can be formulated as : first , we determine the properties of the optimum solution in the following three lemmas .[ lemma : incrs ] under the optimal solution , the transmit powers increase monotonically , i.e. , .assume that the powers do not increase monotonically , i.e. , that we can find two powers such that .the total energy consumed over this duration is .let then , we have , .since , the energy constraint is still satisfied , and thus , the new energy allocation is feasible .we use to replace in the transmission policy , and keep the rest of the rates the same .then , the total number of bits transmitted over the duration becomes where the inequality follows from the fact that is concave in .therefore , the new policy departs more bits by time . keeping the remaining transmission rates the same, the new policy will finish the entire transmission over a shorter duration .thus , the original policy could not be optimal .therefore , the optimal policy must have monotonically increasing powers ( and rates ) .[ lemma : const ] the transmission power / rate remains constant between energy harvests , i.e. , the power / rate only potentially changes when new energy arrives .assume that the transmitter changes its transmission rate between two energy harvesting instances , .denote the rates as , , and the instant when the rate changes as , as shown in fig .[ fig : const_rate ] .now , consider the duration .the total energy consumed during the duration is .let now let us use as the new transmission rate over , and keep the rest of the rates the same .it is easy to check that the energy constraints are satisfied under this new policy , thus this new policy is feasible .on the other hand , the total number of bits departed over this duration under this new policy is where the inequality follows from the fact that is concave in .therefore , the total number of bits departed under the new policy is larger than that under the original policy .if we keep all of the remaining rates the same , the transmission will be completed at an earlier time .this conflicts with the optimality of the original policy .[ lemma : energy ] whenever the transmission rate changes , the energy consumed up to that instant equals the energy harvested up to that instant .  from lemma [ lemma : const ] , we know that the transmission rate can change only at certain energy harvesting instances .assume that the transmission rate changes at , however , the energy consumed by , which is denoted by , is less than .we denote the energy gap by .let us denote the rates before and after by , .now , we can always have two small amounts of perturbations , on the corresponding transmit powers , such that   we also make sure that and are small enough such that , and .if we keep the transmission rates over the rest of the duration the same , under the new transmission policy , the energy allocation will still be feasible .the total number of bits departed over the duration is where the inequality follows from the concavity of in , and the fact that , , as shown in fig .[ fig : concave ] . this conflicts with the optimality of the original allocation .we are now ready to characterize the optimum transmission policy . in order to simplify the expressions ,we let , and let if the transmission completion time lies between and .  based on lemmas [ lemma : incrs ] , [ lemma : const ] and [ lemma : energy ] , we can characterize the optimal policy in the following way . for given energy arrivals , we plot the total amount of harvested energy as a function of , which is a staircase curve as shown in fig . [fig : thm1 ] .the total energy consumed up to time can also be represented as a continuous curve , as shown in fig . [ fig : thm1 ] . in order to satisfy the feasibility constraints on the energy ,energy consumption curve must lie below the energy harvesting curve at all times .based on lemma [ lemma : const ] , we know that the optimal energy consumption curve must be linear between any two consecutive energy harvesting instants , and the slope of the segment corresponds to the transmit power level during that segment .lemma [ lemma : energy ] implies that whenever the slope changes , the energy consumption curve must touch the energy harvesting curve at that energy harvesting instant .therefore , the first linear segment of the energy consumption curve must be one of the lines connecting the origin and any corner point on the energy harvesting curve before .because of the monotonicity property of the power given in lemma [ lemma : incrs ] , among those lines , we should always pick the one with the minimal slope , as shown in fig .[ fig : thm1 ] . otherwise , either the feasibility constraints on the energy will not be satisfied , or the monotonicity property given in lemma 1 will be violated .for example , if we choose the line ending at the corner point at , this will violate the feasibility constraint , as the energy consumption curve will surpass the energy arrival curve .on the other hand , if we choose the line ending at the corner point at , then the monotonicity property in lemma [ lemma : incrs ] will be violated , because in that case , the slope of the following segment would be smaller .these properties must hold similarly for , , , .we also observe that , for given , the optimal transmission policy is the tightest string below the energy harvesting curve connecting the origin and the total harvested energy by time .this is similar to the structure in .we state the structure of the optimal policy formally in the following theorem .[ thm : structure ] for a given , consider a transmission policy with power vector  ] .this policy is optimal if and only if it has the following structure : and for , where is the index of the energy arrival epoch when the power switches to , i.e. , at , switches to .the proof of this theorem is given in appendix [ apdix : thm1 ] .  therefore , we conclude that if the overall transmission duration is known , then the optimal transmission policy is known via theorem [ thm : structure ] . in particular , optimal transmission policy is the one that yields the tightest piecewise linear energy consumption curve that lies under the energy harvesting cure at all times and touches the energy harvesting curve at . on the other hand ,the overall transmission time is what we want to minimize , and we do not know its optimal value up front .consequently , we do not know up front which energy harvests will be utilized . for example , if the number of bits is small , and is large , then , we can empty the data queue before the arrival of , thus , the rest of the energy arrivals are not necessary . therefore , as a first step , we first obtain a good lower bound on the optimal transmission duration .we first illustrate our algorithm through an example in fig .[ fig : proof2 ] .we first compute the minimal energy required to finish the transmission before . we denote it as , and it equals then, we compare it with .if , it implies that we can complete the transmission before the arrival of the first energy harvest , thus is not necessary for the transmission .we allocate evenly to bits , and the duration is the minimum transmission duration . on the other hand ,if , which is the case in the example , the final transmission completion time should be longer than .thus , we move on and compute , , , and find that , and .this means that the total transmission completion time will be larger than and energies , , will surely be utilized .then , we allocate evenly to bits and obtain a constant transmission power , which is the dotted line in the figure .the corresponding transmission duration is .based on our allocation , we know that the final optimal transmission duration must be greater than .this is because , this allocation assumes that all , , are available at the beginning , i.e. , at time , which , in fact , are not .therefore , the actual transmission time will only be larger .thus , is a lower bound for .next , we need to check the feasibility of . observing the figure, we find that is not feasible since it is above the staircase energy harvesting curve for some duration .therefore , we connect all the corner points on the staircase curve before with the origin , and find the line with the minimum slope among those lines .this corresponds to the red solid line in the figure .then , we update with the slope , and the duration for is .we repeat this procedure at and obtain , and continue the procedure until all of the bits are finished .we state our algorithm for the general scenario as follows : first , we compute the amounts of energy required to finish the entire transmission before , , , , respectively , at a constant rate . we denote these as : then , we compare with , and find the smallest such that .we denote this as .if no such exists , we let .now , we assume that we can use to transmit all bits at a constant rate .we allocate the energy evenly to these bits , and the overall transmission time is the solution of and the corresponding constant transmit power is next , we compare with for every . if is smaller than every term , then , maintaining is feasible , and the optimal policy is to transmit at a constant transmission rate with duration , which gives the smallest possible transmission completion time , .otherwise , maintaining is infeasible under the given energy arrival realization .thus , we update i.e. , over the duration , we choose to transmit with power to make sure that the energy consumption is feasible .then , at time , the total number of bits departed is , and the remaining number of bits is .subsequently , with initial number of bits , we start from , and get another lower bound on the overall transmission duration , and repeat the procedure above . through this procedure , we obtain , and the corresponding , until we finish transmitting all of the bits .  based on our allocation algorithm , we know that is optimum up to time , since it corresponds to the minimal slope line passing through the origin and any corner point before . however , the algorithm also implies that the final transmission duration will be larger than .the question then is , whether is still the minimum slope line up to time . if we can prove that is lower than the slopes of the lines passing through the origin and any corner point in  ] , but also between  ] and the corresponding duration vector ]s , we have energy harvested with amounts ] , energies arrive with amounts \times 10^{-2} ] , packets arrive with equal size , as shown in fig .[ fig : example2 ] .we observe that the transmitter changes its transmission power during the transmissions .the first change happens at when energy arrives , and the energy constraint at that instant is satisfied with equality , while the second change happens at when new bits arrive , and the traffic constraint at that time is satisfied with equality .in this paper , we investigated the transmission completion time minimization problem in an energy harvesting communication system .we considered two different scenarios , where in the first scenario , we assume that packets have already arrived and are ready to be transmitted at the transmitter before the transmission starts , and in the second scenario , we assume that packets may arrive during the transmissions .we first analyzed the structural properties of the optimal transmission policy , and then developed an algorithm to obtain a globally optimal off - line scheduling policy , in each scenario .we will prove the necessariness and the sufficiency of the stated structure separately .first , we prove that the optimal policy must have the structure given above .we prove this through contradiction .assume that the optimal policy , which satisfies lemmas [ lemma : incrs ] , [ lemma : const ] and [ lemma : energy ] , does not have the structure given above .specifically , assume that the optimal policy over the duration is the same as the policy described in theorem 1 , however , the transmit power right after , which is , is not the smallest average power possible starting from , i.e. , we can find another , such that based on lemma [ lemma : energy ] , the energy consumed up to is equal to , i.e. , there is no energy remaining at .we consider two possible cases here .the first case is that , as shown in fig .[ fig : thm1_1 ] . under the optimal policy ,the energy required to maintain a transmit power over the duration is .based on ( [ eqn : pn ] ) , this is greater than the total amount of energy harvested during , which is .therefore , this energy allocation under this policy is infeasible .  on the other hand , if , as shown in fig .[ fig : thm1_2 ] , then the total amount of energy harvested over is .from ( [ eqn : pn ] ) , we know thus , under any feasible policy , there must exist a duration , such that the transmit power over this duration is less than .this contradicts with lemma [ lemma : incrs ] .therefore , this policy can not be optimal .next , we prove that if a policy with power vector and duration vector has the structure given above , then , it must be optimal .we prove this through contradiction .we assume that there exists another policy with power vector and duration vector , and the transmission completion time under this policy is smaller .we assume both of the policies are the same over the duration , however , the transmit policies right after , which are and , with durations and , respectively , are different .based on the assumption , we must have .if , from lemma [ lemma : energy ] , we know that the total energy available over is equal to . since , is infeasible over .thus , policy can not be optimal .then , we consider the case when . if , then, the total energy spent over under is greater than , since , and based on lemma [ lemma : incrs ] . if , since the power - rate function is concave , the total number of bits departed over under is greater than that under .thus , policy can not depart bits over , and it can not be optimal .let be the final transmission duration given by the allocation procedure .then , we have . in order to prove that the allocation is optimal , we need to show that the final transmission policy has the structure given in theorem [ thm : structure ] .we first prove that satisfies ( [ eqn : p_min ] ) .then , we can similarly prove that , , satisfy ( [ eqn : p_min ] ) .  we know that if , then it is the minimum possible transmission completion time . we knowthat this transmit policy will satisfy the structural properties in theorem [ thm : structure ] . otherwise , the final optimal transmission time is greater than , and more harvested energy may need to be utilized to transmit the remaining bits . from the allocation procedure ,we know that in order to prove that satisfies ( [ eqn : p_min ] ) , we need to show that   if we keep transmitting with power , then at , the total number of bits departed will be where the inequality follows from the assumption that decreases in .then , ( [ eqn : p1 ] ) guarantees that this is a feasible policy .thus , under the optimal policy , the transmission duration will be upper bounded by , i.e. , which implies if , as shown in fig .[ fig : thm2_1 ] , no future harvested energy is utilized for the transmissions .then , ( [ eqn : pt ] ) guarantees that ( [ eqn : p11 ] ) is satisfied .  if , as shown in fig .[ fig : thm2_2 ] , additional energy harvested after should be utilized to transmit the data .we next prove that ( [ eqn : p11 ] ) still holds through contradiction .assume that there exists with , such that ( [ eqn : p11 ] ) is not satisfied , i.e. , then , combining this with ( [ eqn : t ] ) , we have , which contradicts with the assumption that .thus , ( [ eqn : p11 ] ) holds , satisfies the requirement of the optimal structure in ( [ eqn : p1 ] ) .we can then prove using similar arguments that , , also satisfy the properties of the optimal solution .based on lemma [ thm : structure ] , this procedure gives us the unique optimal policy .first , we prove that for the optimal transmission policy , must satisfy ( [ eqn : r1 ] ) .we prove this through contradiction .if does not satisfy ( [ eqn : r1 ] ) , then , we can always find another , such that first , we assume that .then , if , clearly is not feasible over the duration , because of the energy constraint . if , then , the transmitter can not maintain a transmission rate that is always greater than over , from the energy point of view .this contradicts with lemma [ lemma : incrs2 ] .similarly , if , the `` bottleneck '' is the data constraint . we can prove that is not feasible .thus , must be the smallest feasible rate starting from , as in ( [ eqn : r1 ] ) .we can also prove that , , must have the same structure , in the same way .next , we can prove that any policy has the structure described above is optimal .we can prove this through contradiction .assume that there exists another policy with a shorter transmission completion time .based on lemmas [ lemma : incrs2 ] and [ lemma : energy2 ] , we can prove that this policy could not be feasible .first we prove that obtained through this procedure satisfies ( [ eqn : r1 ] ) .if , i.e. , the constant rate is achievable throughout the transmission , then it is the shortest transmission duration we can get , thus , it is optimal .if , from the procedure , we have we need to prove that considering the policy with a constant power , then , at , the total number of bits departed will be while at , the total energy required will be where the inequality follows from the assumption that decreases in .therefore , maintaining a transmission rate until the last bit departs the system is feasible from both the energy and data arrival points of view .thus , under the optimal policy , the transmission duration will be upper bounded by and , i.e. , which implies if no future harvested energy is utilized for the transmissions , ( [ eqn : pt2 ] ) guarantees that ( [ eqn : r1 ] ) is satisfied .  if , additional energy harvested after should be utilized to transmit the data .we next prove that ( [ eqn : r11 ] ) still holds through contradiction .assume that there exists with , such that ( [ eqn : r11 ] ) is not satisfied , i.e. , then , we have combining this with ( [ eqn : t2 ] ) , we have , which contradicts with the assumption that .thus , ( [ eqn : r11 ] ) holds , satisfies the requirement of the optimal structure in ( [ eqn : r1 ] ) .we can then prove using a similar argument that , , also satisfy the structure of the optimal solution .based on theorem [ thm : struct2 ] , this procedure gives us the unique optimal transmission policy .

Abstract: we consider the optimal packet scheduling problem in a single - user energy harvesting wireless communication system . in this system , both the data packets and the harvested energy are modeled to arrive at the source node randomly . our goal is to adaptively change the transmission rate according to the traffic load and available energy , such that the time by which all packets are delivered is minimized . under a deterministic system setting , we assume that the energy harvesting times and harvested energy amounts are known before the transmission starts . for the data traffic arrivals , we consider two different scenarios . in the first scenario , we assume that all bits have arrived and are ready at the transmitter before the transmission starts . in the second scenario , we consider the case where packets arrive during the transmissions , with known arrival times and sizes . we develop optimal off - line scheduling policies which minimize the time by which all packets are delivered to the destination , under causality constraints on both data and energy arrivals .


elliptic partial differential equations ( pdes ) with discontinuous coefficients and singular source terms , commonly referred to as elliptic interface problems , occur in many applications , including fluid dynamics , material science , electromagnetics , biological systems , and heat or mass transfer . since the pioneer work of peskin in 1977 ,lots of attention has been paid to this field in the past few decades .an elliptic interface problem is formulated as elliptic pdes defined on piecewise - smooth subdomains , which are coupled together via interface conditions , such as given jumps in solution and flux across the domain interface . without the interface , a large variety of convergent numerical schemes are available for solving the elliptic pdes , such as , finite difference method ( fdm ) , finite element method ( fem ) , finite volume method ( fvm ) , wavelet , radial basis functions , meshless and spectral methods . however , for elliptic interface problems , the direct application of the aforementioned schemes can not yield a convergent solution .peskin proposed the immersed boundary method ( ibm ) in order to simulate flow pattern of blood in the heart , in which he approximated the singular sources on the interface . since his work , many numerical methods have been proposed and studied . in 1984 , mayo introduced a second order integral equation approach for poisson s equation and biharmonic equation on irregular domains , in which the solution is extended to a rectangular region by using fredholm integral equations . a fast poisson solver is utilized to solve the resulting fredholm integral equations on a rectangular region .continuous derivatives have been assumed to evaluate the discrete laplacian .this method can deal with jump conditions of \neq 0 ] when the green s function is available . in 1994 , a level set method in combination with the immersed boundary method is proposed by osher and his coworkers in order to compute solutions to incompressible two - phase flow .this level set method is easy to implement , although it is of low convergent rate . in the same year ,immersed interface method ( iim ) was proposed by levique and li , in which the interface conditions are incorporated into the finite difference scheme near the interface to achieve second order accuracy based on a taylor expansion in a local coordinate system .it is a second order interface scheme that does not smear the interface jumps and is used to solve elliptic and parabolic interface problems alike .the resulting linear system is sparse , but not symmetric or positive definite .also , second order derivatives of the interface jumps are needed .the iim is widely used in practice and its extensions can be seen in refs . . for instance , in 2001 , li and ito constructed an iim scheme with resulting linear matrix being diagonally dominate and its symmetric part being negative definite .in addition to the iim , a large class of finite difference based numerical methods have also been proposed for solving elliptic interface problem .the crucial idea of these methods is to incorporate jump conditions into the difference stencils near the interface to maintain high order local truncation error using taylor expansion .numerical schemes based on finite element or finite volume are also developed .finite element schemes are usually constructed by modifying the finite element basis near the interface . a few methods on unfitted mesh have been introduced based on the discontinuous galerkin method using interior penalty technique to handle jump and flux conditions . the weak galerkin finite element method has also been developed for elliptic interface problem .other interesting methods developed in recent decades include the ghost fluid method ( gfm ) by fedkiw , osher and coworkers .the second order convergence of finite difference formalism of interface methods is proved in 2006 by beale and layton for smooth interface , whereas convergence analysis of most elliptic interface schemes is yet to be done .  in the past decade , we have dedicated ourselves to designing accurate and robust numerical schemes for solving elliptic interface problems . in 2006 , the matched interface and boundary ( mib ) method was proposed , motivated by many practical needs , such as optical molecular imaging , nano - electronic devices , , vibration analysis of plates , wave propagation , geodynamics and electrostatic potential in proteins .one important feature of the mib method is its extension of computational domains by using the so called fictitious values , an strategy developed in our earlier methods for handling boundaries and interfaces .as such , standard central finite difference schemes can be employed to discretize differential operators as if there were no interface .another unique feature of the mib method is to repeatedly enforce only the lowest order jump conditions to achieve higher order convergence , which is of critical importance for the robustness of the method to deal with arbitrarily complex interface geometries .higher - order jump conditions must involve higher order derivatives and/or cross derivatives .therefore , to approximate higher order derivatives and/or cross derivatives , one must utilize larger stencils , which is unstable for constructing high - order interface schemes and cumbersome for complex interface geometries .the other distinct feature of the mib method is its dimensional splitting . to enforce a 2d or 3d interface jump condition, we divide the problem into multi 1d ones and resolve a 1d problem at a time , if possible .this approach enables us to come up with a systematic procedure to resolve high dimensional interface problems .finally , based on high order lagrange polynomials , the mib method is of arbitrarily high order in principle .for example , mib schemes up to 16th order accurate have been constructed for simple interface geometries 1d and 2d domains , and sixth - order accurate mib schemes have been developed for complex interfaces in 2d and 3d domains .recently we have constructed an adaptively deformed mesh based mib and a galerkin formulation of mib to improve mib s capability of solving realistic problems .a comparison of the gfm , iim and mib methods can be found in in refs . .the mib method has been used in our earlier works for solving many scientific and engineering problems , such as poisson - boltzmann equation ( pbe ) for describing the electrostatic potential in proteins . to our best knowledge, the mib method is the only method that has demonstrated the second order accuracy for solving poisson or pb equation with realistic protein surfaces with geometric singularities and for solving poisson equation with multiple material interfaces .it can also be used to solve the helmholtz equation for wave scattering and propagation in inhomogeneous media .a fourth order mib scheme for the helmholtz equation with arbitrarily curved interfaces has been proposed by zhao .another example is the geodynamics where the navier - stokes equations with discontinuous viscosity and density is to be solved .et al _ have developed a second order accurate mib method to solve this problem on non - staggered cartesian grids .furthermore , the elastic interface problem in both 2d and 3d domains with arbitrarily complex interface geometry was also addressed by the mib method . in the past few years, the mib method has also been applied to the optical molecular imaging , nano - electronic devices , and vibration analysis of plates .due to the importance and complexity of interface problems , it is still urgent to develop methods that are more accurate , robust and require less computational cost. finite volume method ( fvm ) is known for its ability to better conserve mass and flux in conservation law problems .this property is fundamental for the simulation of many physical models , e.g. , oil recovery simulation and computational fluid dynamics in general .additionally , compared with finite difference method which takes a collocation formulation , fvm is able to deal with solutions with relatively lower regularity . in the present work ,we propose the finite volume formulation of the mib method ( mib - fvm ) .the motivation for the proposed mib - fvm is to inherit the merits of both methods , namely , the capability of the mib method in handling complex interface geometries and the advantage of fvm for conservation laws and low solution regularity .  the rest of this paper is organized as the follows . in section [ theory_and_alg ] , the general theory of the mib based finite volume formulation is briefly discussed .the theoretical formulation and the computational algorithms are given as well . in section [ mib ], we present the strategy for treating complex interface geometries . in section [ numericalstudies ] ,the proposed mib - fvm is validated by benchmark tests , such as 6-petal flower and jigsaw puzzle - like shape in two dimensional ( 2d ) space , and sphere , ellipsoid , cylinder , flower - based cylinder and torus in three dimensional ( 3d ) space .solutions of less regularity ( continuous ) are also tested for the spherical and ellipsoidal interface in 3d .this paper ends with a conclusion .let us consider an open bounded domain with a given interface , which separates the domain into two subdomains , and as illustrated in figure [ interface ] .is divided by interface into two subdomains , and .the mesh is represented by solid lines .the grid points are represented by blue dots.,scaledwidth=50.0% ]  the boundary and interfaces may be non - smooth .the interface can be characterized by a piecewise smooth level - set function , such that . as such, two subdomains can be given by and .the elliptic interface problem can be formulated as where is a piecewise continuous function , is the boundary value , and is a variable coefficient that is discontinuous across the interface . as a result, two jump conditions are required to make the problem well posed     & =  u^+ - u^-   = \phi,~   \forall { \bf x } ~{\rm on } ~\gamma\end{aligned}\ ] ] and   & =  \beta^+ u_n^+ - \beta^- u_n^-  =  \psi,~  \forall { \bf x } ~{\rm on } ~\gamma,\end{aligned}\ ] ] where and denote their limiting value from the side of the interface , and and denote their limiting value from the side of the interface .the derivatives and are evaluated along the outer normal direction on the interface . here and is at least continuous .equations ( [ theoryeq1])-([jumpcondition2 ] ) define the elliptic interface problem to be solved in the present work .   to avoid the time - consuming mesh generation procedure , like the original mib method, we employ the cartesian mesh .we also employ the vertex - centered finite volume method , which means that we associates control volumes and unknowns to vertices .the control volumes are cubes whose centers are located at the grid points and whose sides intercept at the midpoints between grid points , which is illustrated by the left chart of figure [ cv ] .  [ cols="^,^ " , ]   in both cases , the second order derivatives of blow up at the origin , making the solution only continuous in . since the differential form , eq .( [ theoryeq1 ] ) , is of second order for the diffusion term , mathematically , eq .( [ theoryeq1 ] ) is invalid even though the underlying conservation law is still valid .however , it is noticed that the equivalent term in the integral form , eq .( [ integralform1 ] ) , involves only the first order derivative .this reduction of the derivative order is important in dealing with solution which changes so rapidly in space that the spatial derivative does not exist . for these reasons ,the finite volume method is preferred over the finite difference method in solving problems whose solution has less regularity and exhibits local discontinuities .  in case 8(a ) , we choose constant diffusive coefficients and , whereas in case 8(b ) , and are both position dependent . the numerical solutions and the errors of case 8(a ) and case 8(b ) are depicted in figure [ h2sphere_pic ] and figure [ h2ellipsoid_pic ] , respectively . from the numerical results collected in table [ h2sphere_tab ] and table [ h2ellipsoid_tab ], it is seen that the second order accuracy in both cases are essentially obtained .in this work , we introduce the finite volume formulation of matched interface and boundary method ( mib - fvm ) for solving elliptic interface problems arising from modeling material interface in practice . much effort has been taken to develop advanced numerical schemes for such problems in the past few decades .however , challenges still remain in this field .one of the challenge concerns the development of methods with higher order accuracy .another challenge is to develop methods for dealing with complex interface geometries .the matched interface and boundary ( mib ) method has been proved to be able to deal with these challenges . however , since the mib method is based on the collocation formulation , it does not work well for solutions with low regularity .based on the integral form rather than the differential form , the finite volume method ( fvm ) is known for being able to deal with problems with low regularity solutions and better conserve mass and flux .second order finite volume methods have also been formulated in literature for elliptic interface problems. motivated by these successes , we propose the present mib - fvm to take the advantages of mib and fvm .to reduce the computational cost of mesh generation , we utilize the cartesian mesh , although the proposed mib - fvm can be realized on irregular meshes as well .we also employ the vertex - centered fvm , for which the cubic control volumes are associated with the grid point at the center .the control volumes are categorized into regular and irregular types , where special treatments of the mib formulation are needed for the irregular ones in order to maintain the designed accuracy .we study the proposed mib - fvm by a number of classical test cases , including both 2d cases such as 6-petal flower and jigsaw - puzzle like shape , and 3d cases such as sphere , ellipsoid , standard cylinder , flower - based cylinder and torus .the numerical results all demonstrate the second order accuracy of the proposed method .the ability of handling interface with complex geometries and solutions with low regularity ( continuous ) indicates that the proposed mib - fvm combines the advantage of both mib and fvm in meeting numerical challenges .future work will be done to improve the presnt method in dealing with even more complex interface geometries , for example , interface with geometric singularities which are commonly seen in practical applications such as electrostatic analysis of proteins .this work was supported in part by nsf grants iis-1302285 and dms-1160352 , nih grant r01gm-090208 , and msu center for mathematical molecular biosciences initiative .g. guyomarch , c. o. lee , and k. jeon . a discontinuous galerkin method for elliptic interface problems with application to electroporation . in _ commun .numer . meth .engng _ , volume 25 , pages 9911008 , 2009.            j. d. kandilarov .immersed interface method for a reaction - diffusion equation with a moving own concentrated source . in _nma 02 : revised papers from the 5th international conference on numerical methods and applications _ , pages 506513 , london , uk , 2003 .springer - verlag .w. k. liu , y. liu , d. farrell , l. zhang , x. wang , y. fukui , n. patankar , y. zhang , c. bajaj , x. chen , and h. hsu . immersed finite element method and its applications to biological systems ., 195:17221749 , 2006 .

Abstract: the matched interface and boundary ( mib ) method has a proven ability for delivering the second order accuracy in handling elliptic interface problems with arbitrarily complex interface geometries . however , its collocation formulation requires relatively high solution regularity . finite volume method ( fvm ) has its merit in dealing with conservation law problems and its integral formulation works well with relatively low solution regularity . we propose an mib - fvm to take the advantages of both mib and fvm for solving elliptic interface problems . we construct the proposed method on cartesian meshes with vertex - centered control volumes . a large number of numerical experiments are designed to validate the present method in both two dimensional ( 2d ) and three dimensional ( 3d ) domains . it is found that the proposed mib - fvm achieves the second order convergence for elliptic interface problems with complex interface geometries in both and norms .   _ keywords : _  elliptic interface problem ; complex interface geometry ; finite volume method ; matched interface and boundary .


the broadcast channel has predominantly been studied in the context of unicast messaging , where the transmitter wishes to send one private message to each of the receivers ( see for example ) .we refer to this as unicasting .the transmitter may however wish to send different messages to different subsets of receivers .we refer to this as multicasting .the most general multicast structure comprises of messages ( the powerset ) . for are three messages , one required only by the first receiver , one required only by the second receiver , and one required by both receivers .the multicast capacity region for a broadcast channel is the set of -dimensional rate vectors that are achievable . for is the set of achievable rate vectors , where denotes the rate of the common message .one question of interest is , can the multicast capacity region be inferred from the unicast capacity region ?that is , can we always compute the multicast capacity region from the unicast capacity region , i.e. without knowing the structure of the channel ? for certain broadcast channels this is true , although it is not true in general .thus the multicast capacity region provides additional information about the communication limits of the channel beyond that of the unicast capacity region .multicasting has received significant attention in the network - coding literature . in and the maximum rate at which a common message can be sent from a source node through a network of directed noiseless links to a collection of sink nodes ,is shown to equal the minimum - cut of the associated graph . in and the multicast capacity region for one - source - two - sink networksis fully characterized .it is again given by the minimum - cuts of the associated graph . for three or moresinks this is not the case and the problem is open . in this expositionwe shed light on it by characterizing some of the structure for three - sink networks .there is an oddity to multicasting .suppose we have a two - user broadcast channel that can support a rate vector .that is the transmitter can simultaneously deliver one bit of private information to the first receiver , one bit of private information to the second receiver , and one bit of common information to both receivers .an important point to clarify is that there is _ no secrecy requirement _``private '' information sent to the first receiver may or may not be decodable by the second receiver and vice versa .then the channel can also support a rate vector .the transmitter merely uses the common bit to send private information to the first receiver .ofcourse the second receiver is capable of decoding this bit too , but the information is of no interest to it . by symmetrythe achievability of rate vector also implies the achievability of rate vector .there is one more implication in this vein : the achievability of implies the achievability of .the reasoning is similar .the transmitter sends the same information on the two private bits . in this way the first user receives the same private bit as the second user , in addition to the same common bit .thus two common bits have been sent .these three manipulations are summarized in figure [ fig : bc_example ] as _ extremal rays _ stemming from and represent three distinct encoding / decoding operations that can always be performed , regardless of the structure of the broadcast channel . in this sensethey are universal .by time - sharing one can achieve any point in the polytope indicated in figure [ fig : bc_example ] . to summarize : if a rate - vector is achievable , so must be the region illustrated , regardless of the channel .is this set of operations complete ? put in reverse , are there any rate vectors outside the polytope in figure [ fig : bc_example ] that are achievable on for all broadcast channels for which is achievable ?the answer is that there are not there exists a broadcast channel where the rate vector is achievable , but no rate vector outside the polytope in figure [ fig : bc_example ] is .thus for the two - user broadcast channels the three operations discussed form a complete set -they are the only _ distinct universal encoding / decoding operations_.  it is straightforward to generalize these operations to broadcast channels with an arbitrary number of users .consider for example the three user broadcast channel .there are seven messages .suppose a rate vector is achievable ( for example , represents the rate of the message intended for receivers 1 and 3 ) .then for any two subsets of receivers we can perform the operation , and for any two subsets of receivers we can perform the operation .for instance we may swap the first and second receivers private bits for a common bit that is sent to the pair , so that the rate vector is achieved .similarly the rate vector can be achieved by using the first receivers private bit and the bit common to the second and third receivers , to send information common to all three receivers .it can be shown that the number of distinct operations of this form is 15 .that is , if the rate vector is achievable , so is the set of points contained within a 15-edged polytope , which is the generalization to of the polytope in figure [ fig : bc_example ] .again we ask the question , is this set of operations complete ? are there any points outside this 15-edged polytope that are universally achievable on any three - user broadcast channel ?the answer , perhaps surprisingly , is yes. there exists a sixteenth distinct universal encoding / decoding operation .it does not involve a mere relabeling of common and private bits .it enables the rate vector to be achieved .this new operation together with the fifteen trivial ones forms the complete set of distinct universal encoding / decoding operations for .that is , all other rate vectors universally achievable from can be achieved by time sharing between these 16 distinct universal encoding / decoding operations .now we turn to the multiple access channel ( mac ) with users .the mac has also typically been studied in the context of unicast messaging where it s capacity region has in many cases been completely characterized . for multicasting the capacity of the discrete memoryless macis computed in and a conjecture regarding the generalization of this result to an arbitrary number of users is given .let us apply the reasoning we applied above for the broadcast channel , to the mac .consider a two user mac .each transmitter wishes to send a private message of rate to the receiver for .in addition there is a common message of rate that both transmitters share , and desire to be sent to the receiver .suppose for a given mac a rate vector is achievable .then the first transmitter could just label its rate - one bit stream as common information and send it to the transmitter .thus the rate vector is also achievable .by symmetry the second transmitter could do the same so is achievable too .are there any other operations that tradeoff between elements of the rate vector ?is achievable but we will not be interested in this operation as it can be expressed as a linear combination of others . ]the answer is no . for the broadcast channel we could swap common information for private , but not so for the mac .more specifically we can not relabel common information as private , as a common bitstream may require both transmitters have access to it in order for it to be passed to the receiver .a private bitstream assumes only a single transmitter has access to it .the -multicast region for the two - user mac is plotted in figure [ fig : mac_example ] .there are three extremal rays and correspondingly three distinct universal / encoding decoding operations .the first two are stated above and the third consists of merely lowering the common rate so as to arrive at the point .  unlike the broadcast channel , this structure directly generalizes to users . for three usersthere are ten universal encoding / decoding operations .six result from relabeling private information as pairwise .three result from relabeling pairwise as common and the last results from lowering the common rate .thus the multicast capacity region of the multiple access channel has a less intricate structure than that of the broadcast .  in this paperwe characterize the complete set of distinct universal encoding / decoding operations and the associated region of achievable rate vectors , for both the broadcast channel and the mac channel , for .in essence this is a characterization of the universal constraints on the multicast capacity region of these channels .section ii describes the notation we use . in section iiiwe describe the problem in detail .section iv presents the results and section v and vi the proofs .-multicast region for the broadcast channel , . ]-multicast region for the multiple access channel , . ]we briefly describe some of the notation that will be used . typically and will be used to denote subsets of .for example we may have , which would imply .rates in bold font represent tuples , for example we may have .elements of time series are indicated by a index in parentheses following the variable , for example .an entire time series is represented by bold font , for example  ] .a -universal encoding / decoding operation is called ` distinct ' if the vector can not be expressed as positive linear combination of vectors for which there exist -universal encoding / decoding operations for .the ( rays associated with the ) normalized difference vectors corresponding to distinct -universal encoding / decoding operations are called ` extremal rays ' .  by positive linear combinationwe mean a weighted linear sum with non - negative coefficients .it will be evident later that the universal encoding / decoding operation given in example [ exm : universal encoding ] is distinct and thus is an extremal ray . by symmetry is also an extremal ray .note distinctness does not imply uniqueness the universal encoding / decoding operation that moves from rate vector to rate vector is also classified as distinct , but it has the same normalized difference vector .an example of a universal encoding / decoding operation that is not distinct is one that moves from rate vector to rate vector .denote the corresponding normalized difference vector is .the universal encoding / decoding operations that achieve this shift correspond to time - sharing between two operations , one with normalized difference vector , the other with normalized difference vector .indeed we have   we now give a formal definition of the region alluded to in figure [ fig : bc_example ] .let   be a parameter .the ` -multicast region ' is the intersection of the capacity regions of all broadcast channels for which the rate vector is achievable , i.e.   see figures [ fig : bc_example ] for examples of this region .as the problem setup for the multiple access channel is entirely analogous to the aforementioned setup for the broadcast channel , we do not explicitly describe it .an example of the -multicast region is given in figure [ fig : mac_example ]  the aim of this paper is to characterize the -multicast cones for both the broadcast and multiple access channels .[ fig : matricesl2 ] \quad\quad { \bf h}_{bc,2 } = \left[\begin{array}{rrr }  1 & 0 & -1 \\  0 & 1 & -1 \\  -1 & -1 & 1 \\ \end{array}\right]\ ] ] \quad\quad { \bf h}_{mac,2 } = \left [ \begin{array}{rrr }  1 & 0 & -1 \\  0 & 1 & -1 \\-1 & -1 & 1 \\ \end{array}\right]\ ] ]  [ fig : matricesl3 ]  \ ] ]  \ ] ]  \ ] ]  \ ] ]  [ thm : main_result_bc ] for the -multicast region of the broadcast channel is the set of all satisfying where is given in figure [ fig : matricesl3 ] .this region is a polytope , characterized by the cone .we refer to this cone as the multicast cone. the sixteen extremal rays of this cone are given by the columns of the matrix in figure [ fig : matricesl3 ] .thus there are 16 distinct universal encoding / decoding operations for .the -multicast region for the broadcast channel for is illustrated in figure [ fig : bc_example ] . for are 3 distinct universal encoding / decoding operations .the and matrices are given in figure [ fig : matricesl2 ] .the columns of are the normal vectors to the three hyperplanes bounding the region .the columns of are the three extremal rays ( see figure [ fig : bc_example ] ) .[ thm : main_result_mac ] for the -multicast region of the multiple access channel is the set of all satisfying where is given in figure [ fig : matricesl3 ] .this region is also a polytope characterized by the cone .the 10 extremal rays of this cone are given by the columns of the matrix in figure [ fig : matricesl3 ] .thus there are 10 distinct universal encoding / decoding operations for .the -multicast region for the mac for is illustrated in figure [ fig : mac_example ] .there are 3 distinct universal encoding / decoding operations .the and matrices are given in figure [ fig : matricesl2 ] .an alternative interpretation of theorem [ thm : main_result_bc ] is the following ( the same interpretation applies for [ thm : main_result_mac ] ) . for notational simplicitywe denote the capacity region of an arbitrary broadcast channel by .let   is the rate vector lying on the boundary of the capacity region in the direction of .let   is the halfspace of all rate vectors lying underneath the hyperplane .the region is convex and thus we can characterize it by its support function , i.e.   however this is not the minimal dual representation of .let   the multicast capacity region of an any broadcast channel with three receivers can be expressed as   if and only if   this says the following : when computing the multicast capacity region of a broadcast channel by maximizing the weighted sum - rate , the smallest set that one need vary the weighting coefficients over is .put another way , the normal vector to any point on the boundary of the multicast capacity region is always contained in the set .see figure [ fig : valid_region ].   of the broadcast channel capacity region satisfies .( a ) a capacity region that can not occur .( b ) a capacity region that can occur . ]the direct part of the proof consists of showing that for any broadcast channel , if a rate vector is achievable then all rate vectors in the region given by equation ( 1 ) are achievable .this establishes that the -multicast region is ` at least as large ' as the region given by equation ( 1 ) .the converse part of the proof consists of illustrating , for each , a broadcast channel for which no rate vector outside the region given by equation ( 1 ) is achievable .this establishes that the -multicast region is ` at least as small ' as the region given by equation ( 1 ) .we start with the direct part .for notational simplicity we drop the broadcast channel ( _ bc _ ) subscript .suppose that is achievable for a particular broadcast channel .we show that any rate - vector satisfying   for is also achievable .we then show that this region is precisely the one given in equation ( 1 ) .let denote the element of and denote the column of . to show that any rate - vector satisfying equation ( 2 ) is achievable , we show that each of the 16 rate - vectors given by   are achievable where by time sharing between these vectors the entire boundary region achieved and hence any point within it ( i.e. satisfying equation ( 2 ) ) can also be achieved .let correspond to the binary message vector and estimate of the message vector , respectively , that the transmitter wishes to send at rate vector .we illustrate the achievability of equation ( 3 ) for .  to universally encode for , assume without loss of generality that . in what followswe ignore rounding effects as it will be clear that in the limit they are negligible .set \\ w_2^n & = [ m_{12}(1),\dots , m_{12}(nr_1^ * ) , m_{2}(1),\dots , m_{2}(nr_2^*-nr_1^ * ) ] \\ w_{12}^n & = [ m_{12}(nr_1^*+1),\dots , m_{12}(nr_{1}^*+nr_{12}^*)]\end{aligned}\ ] ]  in words , the information common to receivers 1 and 2 is split into two parts . the first part is replicated and sent separately down both receiver 1 and receiver 2 s private channels . the second part is sent down the channel common to both receivers . as receiver 2 s private channel can accommodate a higher bit - rate than receiver 1 s , there is some bandwidth left over .this is allocated to sending some of receiver 2 s private information .  for all other subsets of set and .universal decoding is straightforward .the first receiver sets \\{ \hat m}_{1,13}^n & = { \hat w}_{13}^n \\ { \hat m}_{1,123}^n & = { \hat w}_{123}^n\end{aligned}\ ] ]  and in this way successfully recovers its message , as the achievability of implies that was decoded correctly .the second receiver sets \\ { \hat m}_{2,12}^n & = [ { \hat w}_2(1),\dots,{\hat w}_2(nr_1^ * ) , { \hat w}_{12}^n ] \\ { \hat m}_{2,23}^n & = { \hat w}_{23}^n \\ { \hat m}_{2,123}^n & = { \hat w}_{123}^n\end{aligned}\ ] ]  and is similarly successful in decoding .the third receivers sets for all of its messages .then we have achieved a rate vector of r_1^ * \\ & = { \bf r}^ * + { \bf h}_{bc,3}(3)\delta_3.\end{aligned}\ ] ] with .the universal encoding and decoding procedures for all other are similar and follow from the structure of the columns of the matrix .universal encoding and decoding for is different .assume without loss of generality that .to encode , set for and \\ w_{13}^n & = [ m_{123}(nr_{12}^*+1),\dots , m_{123}(2nr_{12}^*),m_{13}(1),m_{13}(nr_{13}-nr_{12 } ) ] \\ w_{23}^n & = [ m_{123}(1)\oplus m_{123}(nr_{12}^*+1),\dots , m_{123}(nr_{12}^ * ) \oplus m_{123}(2nr_{12}^ * ) , \\& \quad\quad\quad m_{23}(1),\dots , m_{23}(nr_{23}-nr_{12 } ) ] \\ w_{123}^n & = [ m_{123}^n(2nr_{12}^*),\dots , m_{123}^n(2nr_{12}^*+nr_{123}^*)]\end{aligned}\ ] ]  in words , the information common to all receivers is split into three streams .the first and second are sent at rate using the three pairwise links .the third stream is sent at rate across the link common to all receivers .the first receiver decodes by setting and \\ m_{123}^n & = [ w_{12}^n , w_{13}^n , w_{123}^n].\end{aligned}\ ] ]  the second receiver decodes by setting and \\ m_{123}^n & = [ w_{12}^n , w_{12}^n\oplus w_{23}^n , w_{123}^n].\end{aligned}\ ] ]  the third receiver decodes by setting and \\ m_{23}^n & = [ w_{23}(nr_{12}^*+1),\dots , w_{23}(nr_{23}^ * ) ] \\ m_{123}^n & = [ w_{13}^n , w_{13}^n\oplus w_{23}^n , w_{123}^n].\end{aligned}\ ] ]  then we have achieved a rate vector of r_{12}^ * \\ & = { \bf r}^ * + { \bf h}_{bc,3}(16)\delta_{16}.\end{aligned}\ ] ] with . thus the 16 rate vectors satisfying equation ( 3 ) are achievable and by time sharing between them , all rate vectors in the region given by equation ( 2 ) are achievable .  it remains to show that this region is equivalent to the one in equation ( 1 ) , i.e. that for any   on the left is the characterization of the polytope in terms of the hyperplanes bounding it . on the right is the dual characterization in terms of the edges of the polytope ( 1-dimensional facets ) .this equivalence can be demonstrated using computer software such as _ polymake_.   to establish the converse we now present , for each , a particular ( deterministic ) broadcast channel and show its capacity region is equal to ( 1 ) .let the input alphabet with the channel input \ ] ]  so that each , and let for with \\y_2(i ) & = [ x_2(i),x_{12}(i),x_{23}(i),x_{123}(i ) ] \\ y_3(i ) & = [ x_3(i),x_{13}(i),x_{23}(i),x_{123}(i)].\end{aligned}\ ] ]    see figure [ fig : specific_bc ] for an illustration of the channel .suppose the channel is used times .the messages to be transmitted are and mutually independent . denote the length- vector of channel inputs by and the length- vectors of channel outputs by and  denote the column of .we wish to show   for . before thiswe introduce some notation .suppose is a collection of subsets of , for example .the collection should be thought as the indices of a subset of the seven channel links ( see figure [ fig : specific_bc ] ) , for example corresponds to two links , the private one from to and the common one from to . by denote the indices of the messages intended for those receivers cut by .for example if then all links to the first receiver are cut , but not all links to the second or the third . as the first receiveris sent the messages and , we have . as another examplelet .then all links to both the second and third receivers are cut and . as a final example if no receivers are completely cut , and thus .let and be three collections of subsets of such that either , or .then   this lemma is a generalization of the cutset bounds to multiple subsets of cuts .indeed if we set and we are left with   which are precisely the cutset bounds .  where the third step follows from lemma [ lem : habc_expansion ] in the appendix and the fourth from the requirement ( fano s inequality ) and lemma [ lem : igreaterthanh ] in the appendix .applying lemma 5.1 to the sets of indices in table 1 establishes equation ( [ eqn : bc_region ] ) for columns of .unfortunately for column the condition that either , or must hold , is violated .consequently the 11th converse bound is established in a different fashion .let be defined by the 11th row of table 1 . then    where the fourth step follows from the requirement ( fano s inequality ) , the sixth from lemma [ lem : zeroint condsets ] and the seventh from the independence of the messages .[ cols="^,^,^,^",options="header " , ]the direct part of this proof is entirely analogous to the direct part for the broadcast channel .this establishes the universal achievability of the -multicast region .the converse part is different . for each we present a sequence of channels .the limiting intersection of the capacity regions of these channels is the region in equation ( [ eqn : mac_region ] ) .the capacity regions of these channels are not precisely computed , but only outer bounded in a manner sufficient to establish their limiting intersection .   as this part of the proof is trivial and entirely analogous section 5.1we only provide a sketch .in essence we need to establish that each of the columns of are achievable in the sense of section 5.1 .the first column is achieved by transmitting additional bits on the channel , the second column is achieved by transmitting additional bits on the channel , the third column is achieved by transmitting additional bits on the channel , and so on .the last column is achieved by lowering the rate of the message .   for each we present a sequence of deterministic channels with capacity region tending to the region in equation ( [ eqn : mac_region ] ) .the capacity regions of these channels are not explicitly computed , only outer bounded , but we show the limiting outer bound is tight .the sequence is parameterized by the integer .let be given and assume its elements are rational .denote their numerators and denominators by and , for so that .let .the channel is defined as follows .see figure [ fig : coor_channel ] for a pictorial representation .every time steps the channel takes in a triple of inputs and outputs one symbol .the input alphabet is where   the output alphabet is   where is an output symbol that can be thought of as an erasure .the channel thus decomposes into one with inputs and 7 outputs .the outputs at time are related deterministically to the inputs at time via    the input streams thus consist of blocks of bits .the output streams match their associated input streams . the output stream matches its associated input streams if and only if the input streams match at each bit , otherwise the erasure symbol is outputted . likewise for the other output streams . for this reasonthe boxes inside the channel in figure [ fig : channel_mac ] are labeled coordination channel. see figure [ fig : coor_channel ] for a pictorial example of one such coordination channel .the idea of the coordination channels is that in the limit of large , they only let common information through .this should be intuitive from their definition and from the figure .    .( b ) a coordination channel for . ]we now bound the capacity region of this channel .it is clear that we can further decompose the channel into seven parallel channels , one linking and , one linking and , one linking and , one linking and , one linking and , one linking and , and one linking and .the capacity region of the channel in question is thus the minkowski sum of the capacity regions of these seven channels .denote these seven capacity regions by for .then the capacity region of our channel is given by   where sigma denotes the minkowski sum .in particular we wish to compute the limiting intersection of these regions   1 .the region is the set of all satisfying and for , 2 .the region is the set of all satisfying and for , 3 .the region is the set of all satisfying and for , 4 .the region is the set of all satisfying and for , 5 .the region is the set of all satisfying and for , 6 .the region is the set of all satisfying and for , 7 .the region is the set of all satisfying and for .the first three regions are trivial .we establish the fourth .the messages are uniformly distributed on and mutually independent for fixed .denote the -length output sequence by .by fano s inequality we must have with as in order for the error probability to be made arbitrarily small .thus by the mutual independence of the messages we have   assume for simplicity that where is an integer .we write $ ] where represents the block of symbols in .similarly represents the block of symbols in .we also use the shorthand .we proceed to show that is sufficiently small .  from the channel definition we have   using this expression and the conditional independence of and given we have    the first term can be upper bounded by 1 ( as for all ) .the second term can also be upper bounded by 1 . to see this , maximize first over the distribution and then over the distribution , \log \left [ \max_{x}p({\bf x}_{1,12}^i = x|{\bf w}={\bf w } ) \right ] \\ & \quad\quad \le 1\end{aligned}\ ] ]  likewise the third term can be upper bounded by 1 . thus putting this all together we have   and so   then by letting we have for . from the structure of the coordination channelit is clear that we can achieve points and . by time - sharingwe can achieve all points in the region .conversely from fano s inequality we have   where as .this establishes the fourth component of the lemma .the remaining components are established in the same manner .we omit the details .it remains to show that the region , corresponds to the region in equation ( [ eqn : mac_region ] ) .[ lem : igreaterthanh ] let and be three sets of random variables satisfying at least one of the properties , or .let be a random variable that satisfies for . then       [ lem : zeroint condsets ] let be a set of mutually independent r.vs .let and be three subsets of these r.v.s with the property . then for any r.v .    where the third step follows from lemma [ lem : habc_expansion ] , the fourth from a set expansion made possible by the mutual independence of the underlying r.v.s , the fifth from the property , the sixth by a set relationship , and the eighth by lemma [ lem : igreaterthanh ] .the second step follows from the containment property .the first term in the fourth step follows by applying lemma [ lem : igreaterthanw ] with , , and , the second term by applying the same lemma with , , and .the third term in the third and fourth steps are equal by the containment property .

Abstract: much of the existing work on the broadcast channel focuses only on the sending of private messages . in this work we examine the scenario where the sender also wishes to transmit common messages to subsets of receivers . for an -user broadcast channel there are subsets of receivers and correspondingly independent messages . the set of achievable rates for this channel is a -dimensional region . there are fundamental constraints on the geometry of this region . for example , observe that if the transmitter is able to simultaneously send rate - one private messages , errorfree to all receivers , then by sending the same information in each message , it must be able to send a single rate - one common message , errorfree to all receivers . this swapping of private and common messages illustrates that for any broadcast channel , the inclusion of a point in the achievable rate region implies the achievability of a set of other points that are not merely componentwise less than . we formerly define this set and characterize it for and . whereas for all the points in the set arise only from operations relating to swapping private and common messages , for a form of network coding is required .


the problem of managing operations of a telephone call center in an efficient way has a long history in the area of operational research and is a topic of current research in various disciplines ( see e.g. or for extensive overviews ) .from the modeling point of view they can be viewed as queuing systems .such a queuing model can be described by a corresponding continuous time markov chain ( ctmc ) whose steady - state distribution can be easily determined , either analytically - with the erlang - c formula for the simplest m / m / n model or with the erlang - a formula for its version augmented with exponential patience time as proposed in ; or numerically for more complicated models ( as in or recently ) . however , as real call centers are time inhomogenous , with varying arrival rates and changing number of servers - scheduled to meet the forecasted demand and in order to provide break time , stationary models can not be applied directly .it is , therefore , common to use approximations , assuming the system being pointwise stationary .examples of such well established methods can be found e.g. in , or in .unfortunately , stationary approximations are in many cases not adequate .for example , compared them with simulations based on real inbound call center data , with the conclusion that due to the nonstationarity only some of the performance measures can be estimated with satisfactory accuracy .ingolfsson in compared them with an inherently transient model and found their results significantly inaccurate or even entirely unreliable . despite this , their widespread use is commonly justified by simple implementation and low computational costs .many authors proposed to use simulation , which can achieve any desired accuracy .however , in order to achieve acceptable precision , very long computational times are needed , which makes it often impracticable for common applications like schedule planning .an alternative approach , which is very effective in terms of the accuracy of the model , is to analyze transient ctmc using numerical methods , solving effectively their corresponding system of _ ordinary differential equations _( ode s ) as proposed in , or by the author in .other , less computationally intensive , analytical methods that can approximate such nonstationary systems more accurately than stationary models are _ closure approximations _ and _ fluid and diffusion approximations _ , discussed e.g. in , , and or , for the direct comparison of some examples of such methods with the numerical methods and stationary approximations , in .although there is a number of papers dealing with the phenomena of customer balking and abandonment in multiserver queues ( e.g. ,,, or recently ) , they concentrate on stationary models or approximations . to the best of our knowledge , an inherently transient ctmc model dealing with both balking and abandonment of a call center , has never been investigated .the main objective of this work is to model such non - stationary systems , using transient analysis of corresponding ctmc , in a reliable and precise way , with computational efficiency enabling its use for practical applications in particular , as a much more accurate replacement to the erlang - c and erlang - a formulas , used by practitioners for quantitative call center management .  in this paperwe model an inbound telephone call center with balking and abandonment , i.e. the customer may not stay in the queue once realizing he is put on hold , or abandon the queue if the waiting time is too long , extending the nonstationary m / m / n queuing model analyzed by the author in .the paper is structured as follows . in the next section the model and the basic notationare introduced .section 3 reviews the proposed multi - step uniformization algorithm with steady - state detection and section 4 presents the results of numerical experiments .the paper ends with a summary of results , conclusions and proposals for future research .we propose a following model of a call centre : the analyzed period is finite ( e.g. one working day ) with the system starting empty . the state variable represents total number of service requests ( served / waiting calls ) in the system at time .the size of the system , which represents the number of non empty possible states , is finite , equal to = number of identical servers ( agents ) plus = capacity of the queue , with corresponding discrete state space , .customers arrive according to an inhomogenous poisson process with rate , the service time is i.i.d . exponentially distributed with rate .the load can be bigger than 1 .service requests that are not served immediately can leave the system ( hang up or balk ) with probability 1- , otherwise , after joining the queue , they abandon after reaching their _ patience time_. the patience times are independent and identically exponentially distributed with mean .queued requests are fcfs served .all of this is modeled via the state transition rates of a ctmc which is described by _ infinitesimal generator matrix _ and the _ initial state probability vector _ , where the time dependent value is the rate at which the state changes to the state and represents the rate for the event of staying in the same state .because is a birth - and - death process , it can be described by following state dependent birth and death rates :    similar to the model in .the transient distribution at time t for a given time dependent generator matrix can be calculated using kolmogorovs forward equations : where the vector  ] the value of each its state has to be computed with an error less than .let us further assume being the error after computing some .then :   as the error bound of steady state approximation is , in case the steady state is reached , absolute and independent of the error of the previous steps , we can set the convergence threshold dependent rather on the actual total error bound than the error for the single step ( as proposed e.g. by ) .it allows , consequently , to trade the error bounds of steps for higher convergence thresholds while still within the global error bound for the whole solution .then , assuming the system at time  to satisfy for each , $ ] we have to :   test the implementation the following model has been used : a service system ( call center ) working for time = 24h and starting empty .the arrival rate changes sinusoidal with two peaks and is divided into 288 ( 5min ) periods with constant averaged rates , same as the first example in .the service rate and number of servers are constant ( , ) , the arrival rate varies in time - ( the load varying between 0.65 and 1.05 as shown in figure [ load ] ) .the probability of a customer immediately leaving when not served immediately is 0.03 .the mean value of patience time is equal to 4min .the capacity of the queue is constant and chosen so that for all times the probability of the system being in the state is less than for all tested system sizes .    to evaluate the impact of the proposed steady - state detection algorithm , models of 5 different sizes have been at first calculated using unmodified uniformization algorithm with an error step corresponding to the total error bound .[ table_ssd_taylor ]  the detailed results of computation times are in table [ table_ssd_taylor ] .all experiments were performed on a 1.7ghz pc under 64bit linux os with a processor supporting vector operations in both : avx with 256bit vectors ( 4 double or 8 float operations simultaneously ) and the older sse instruction set with 128bit vector operations ( an intel i5 - 3317u with cpu throttling disabled via kernel scaling governor ) , compiled with gnu gcc compiler .all measurements use standard unix _time.h / clock ( ) _ function - returning cpu time .all times are in milliseconds .load ,s=1000 q=200  the impact of reduced computational effort due to steady - state detection for some chosen total error bounds ( between and ) , with corresponding steady - state detection thresholds , is illustrated for the system of size 1200 in figure [ ssd_abdmnt_65_105_taylor_1000_200 ] .load , s=1000 q=200    load , s=1000 q=200  figure [ ssd_abdmnt_65_105_ess_taylor_1000_200 ] shows the expected state of the system , derived from the calculated probability vector as :  \ ] ]  figure [ ssd_abdmnt_65_105err_taylor_1000_200 ] shows its relative error for different steady - state detection thresholds .the reference for the error estimate has been calculated with .load , s=1000 q=200  figure [ ssd_abdmnt_65_105_sl0_taylor_1000_200 ] shows the probability for an incoming service request to be served immediately ( with no waiting time ) .[ table_gamma_eta_65_105 ]  table [ table_gamma_eta_65_105 ] shows computation times and maximal values of  probability of the system being in the state , for different values of : queue length , probability of the customer entering the queue despite not being served immediately and mean value of patience time .the difference compared to the corresponding value for and = 4min from the table [ table_ssd_taylor ] ( equal to 468ms ) is not only both due to the bigger size of the system and higher uniformization rate resulting from higher queue length , but also to some extent due to the higher variability of the system state , resulting in fewer steps where the steady - state could be detected within the respective threshold .to illustrate this effect , we repeated the experiment with the load variability reduced to only i.e. with the arrival rate .the results corresponding to the cases from the table [ table_gamma_eta_65_105 ] are shown in the table [ table_gamma_eta_95_105 ] .[ table_gamma_eta_95_105 ]in this paper we showed that the uniformization with steady - state detection can be used in a very effective way to evaluate transient behavior of multiserver queues . applied to the modeling of the call center schedules, it allows calculation of transient system states for systems of any , possible in practical applications , size in a very short time , in a numerically stable way , with very high precision , using relatively common and inexpensive cpu .it can , therefore , be used for schedule planning based on available forecasts , as described in .the presented method can be extended in several directions .one could be , in regard to call center modeling , to automatically optimize the model size ( queue length ) with significant impact on the computational efficiency .another could be to use known periodicity of traffic forecasts to divide total error bound in between known times of the day , bounded by the points of time when the system will reach a steady state , than for the whole modeled period .

Abstract: this paper considers a nonstationary multiserver queuing model with abandonment and balking for inbound call centers . we present a continuous time markov chain ( ctmc ) model which captures the important characteristics of an inbound call center and obtain a numerical solution for its transient state probabilities using uniformization method with steady - state detection .   keywords : call center , transient , markov processes , numerical methods , uniformization , abandonment , balking