new families of binary sequences with optimal three level autocorrelation in this correspondence we give several new families of binary sequences of period n with optimal three level autocorrelation where n spl equiv 2 mod 4 these sequences are either balanced or almost balanced our construction is based on cyclotomy

the stability of two station multitype fluid networks this paper studies the fluid models of two station multiclass queueing networks with deterministic routing a fluid model is globally stable if the fluid network eventually empties under each nonidling dispatch policy we explicitly characterize the global stability region in terms of the arrival and service rates we show that the global stability region is defined by the nominal workload conditions and the virtual workload conditions and we introduce two intuitively appealing phenomena virtual stations and push starts that explain the virtual workload conditions when any of the workload conditions is violated we construct a fluid solution that cycles to infinity showing that the fluid network is unstable when all the workload conditions are satisfied we solve a network flow problem to find the coefficients of a piecewise linear lyapunov function the lyapunov function decreases to zero proving that the fluid level eventually reaches zero under any nonidling dispatch policy under certain assumptions on the inter arrival and service time distributions a queueing network is stable or positive harris recurrent if the corresponding fluid network is stable thus the workload conditions are sufficient to ensure the global stability of two station multiclass queueing networks with deterministic routing

heavy traffic optimality of a stochastic network under utility maximizing resource allocation we study a stochastic network that consists of a set of servers processing multiple classes of jobs each class of jobs requires a concurrent occupancy of several servers while being processed and each server is shared among the job classes in a head of the line processor sharing mechanism the allocation of the service capacities is a real time control mechanism in each network state the resource allocation is the solution to an optimization problem that maximizes a general utility function whereas this resource allocation optimizes in a greedy fashion with respect to each state we establish its asymptotic optimality in terms of a deriving the fluid and diffusion limits of the network under this allocation scheme and b identifying a cost function that is minimized in the diffusion limit along with a characterization of the so called fixed point state of the network

the linear complexity of binary sequences with optimal autocorrelation binary sequences with optimal autocorrelation are needed in many applications two constructions of binary sequences with optimal autocorrelation of period n 0 mod 4 are investigated the two constructions are powerful and generic in the sense that many classes of binary sequences with optimal autocorrelation could be obtained from binary sequences with ideal autocorrelation general results on the minimal polynomials of these binary sequences are derived based on the results both the linear complexities and the minimal polynomials are determined

diffusion limit of fair resource control stationarity and interchange of limits we study a resource sharing network where each job requires the concurrent occupancy of a subset of links servers resources and each link s capacity is shared among job classes that require its service the real time allocation of the service capacity among job classes is determined by the so called proportional fair scheme which allocates the capacity among job classes taking into account the queue lengths and the shadow prices of link capacity we show that the usual traffic condition is necessary and sufficient for the diffusion limit to have a stationary distribution we also establish the uniform stability of the prelimit networks and hence the existence of their stationary distributions to justify the interchange of two limits the limit in time and limit in diffusion scaling we identify a bounded workload condition and show it is a sufficient condition to justify the interchange for the stationary distributions and their moments this last result is essential for the validity of the diffu

limit theorems for markovian bandwidth sharing networks with rate constraints bandwidth sharing networks provide a natural modeling framework for describing the dynamic flow level interaction among elastic data transfers in computer and communication systems and can be used to develop traffic pricing charging mechanisms at the same time such models are exciting from an operations research perspective because their analysis requires techniques from stochastic modeling and optimization in this paper we develop a framework to approximate bandwidth sharing networks under the assumption that the number of users as well as the capacities of the system are large and the assumption that the traffic that each user is allowed to submit is bounded above by some rate which is standard in practice we also assume that customers on each route in the network abandon according to exponential patience times under markovian assumptions we develop fluid and diffusion approximations which are quite tractable for most parameter combinations the invariant distribution is multivariate normal with mean and diffusion coefficients that can be computed in polynomial time as a function of the size of the network

new designs for signal sets with low cross correlation balance property and large linear span gf p case new designs for families of sequences over gf p with low cross correlation balance property and large linear span are presented the key idea of the new designs is to use short p ary sequences of period spl upsi with the two level autocorrelation function together with the interleaved structure to construct a set of long sequences with the desired properties the resulting sequences are interleaved sequences of period spl upsi sup 2 there are spl upsi cyclically shift distinct sequences in each family the maximal correlation value is 2 spl upsi 3 which is optimal with respect to the welch bound each sequence in the family is balanced and has large linear span in particular for binary case cross out of phase autocorrelation values belong to the set 1 spl upsi spl upsi 2 2 spl upsi 3 2 spl upsi 1 any sequence where the short sequences are quadratic residue sequences achieves the maximal linear span it is shown that some families of these sequences can be implemented efficiently in both hardware and software

heavy traffic limits for queues with many exponential servers two different kinds of heavy traffic limit theorems have been proved for s server queues the first kind involves a sequence of queueing systems having a fixed number of servers with an associated sequence of traffic intensities that converges to the critical value of one from below the second kind which is often not thought of as heavy traffic involves a sequence of queueing systems in which the associated sequences of arrival rates and numbers of servers go to infinity while the service time distributions and the traffic intensities remain fixed with the traffic intensities being less than the critical value of one in each case the sequence of random variables depicting the steady state number of customers waiting or being served diverges to infinity but converges to a nondegenerate limit after appropriate normalization however in an important respect neither procedure adequately represents a typical queueing system in practice because in the heavy traffic limit an arriving customer is either a

asymptotic stationarity of queues in series and the heavy traffic approximation one deals with a system of queues in series that is to say a number of queues through which a unit passes in turn spending a waiting time including service in any particular one and proceeding to the next succeeding queue immediately the service time is completed the first main result deals with the asymptotic behaviour of the vector of waiting times of a unit in each queue and of the time which a unit spends in the tandem queue the second main result deals with the heavy traffic approximation of the mentioned characteristics

validity of heavy traffic steady state approximations in multiclass queueing networks the case of queue ratio disciplines a class of stochastic processes known as semi martingale reflecting brownian motions srbms is often used to approximate the dynamics of heavily loaded queueing networks in two influential papers bramson bramson m 1998 state space collapse with applications to heavy traffic limits for multiclass queueing networks queueing systems 30 89 148 and williams williams rj 1998b diffusion approximations for open multiclass queueuing networks sufficient conditions involving state space collapse queueing systems 30 27 88 laid out a general and structured approach for proving the validity of such heavy traffic approximations in which an srbm is obtained as a diffusion limit from a sequence of suitably normalized workload processes however for multiclass networks it is still not known in general whether the steady state distribution of the srbm provides a valid approximation for the steady state distribution of the original network in this paper we study the case of queue ratio disciplines and provide a set of sufficient conditions under which the above question can be answered in the affirmative in addition to standard assumptions made in the literature towards the stability of the pre and post limit processes and the existence of diffusion limits we add a requirement that solutions to the fluid model are attracted to the invariant manifold at a linear rate for the special case of static priority networks such linear attraction is known to hold under certain conditions on the network primitives the analysis elucidates interesting connections between stability of the pre and post limit processes their respective fluid models and state space collapse and identifies the respective roles played by all of the above in establishing validity of heavy traffic steady state approximations

fluid limits for bandwidth sharing networks with rate constraints bandwidth sharing networks as introduced by roberts and massoulie roberts jw massoulie l 1998 bandwidth sharing and admission control for elastic traffic proc itc specialist seminar yokohama japan massoulie and roberts massoulie l roberts jw 1999 bandwidth sharing objectives and algorithms proc ieee infocom books in statistics new york 1395 1403 model the dynamic interaction among an evolving population of elastic flows competing for several links with policies based on optimization procedures such models are of interest both from a queueing theory and operations research perspective in the present paper we focus on bandwidth sharing networks with capacities and arrival rates of a large order of magnitude compared to transfer rates of individual flows this regime is standard in practice in particular we extend previous work by reed and zwart reed j zwart b 2010 limit theorems for bandwidth sharing networks with rate constraints revised preprint http people stern nyu e

new construction for families of binary sequences with optimal correlation properties we present a construction in a closed form for an optimal family of 2 sup m binary sequences of period 2 sup 2m 1 with respect to welch s 1974 bound whenever there exists a balanced binary sequence of period 2 sup m 1 with ideal autocorrelation property using the trace function this construction enables us to reinterpret a small set of kasami and no 1988 sequences as a family constructed from m sequences new optimal families of binary sequences are constructed from the legendre sequences of mersenne prime period hall s sextic residue sequences and miscellaneous sequences of unknown type in addition we enumerate the number of distinct families of binary sequences which are constructed from a given binary sequence by this method

gmw sequences corresp the difference set design of gordon mills and welch gmw is adapted for use as a pseudorandom number generator statistical properties of the generated binary sequences including periodic correlation linear span and k tuple statistics are derived one mechanization of a gmw sequence generator is suggested and the number of sequences that can be generated with a fixed number of shift register stages and read only memory rom size is evaluated

new construction for binary sequences of period p sup m 1 with optimal autocorrelation using z 1 sup d az sup d b we present a construction for binary sequences s t of period n p sup m 1 for an odd prime p based on the polynomial z 1 sup d az sup d b and discuss them in some cases of parameters p m d a and b we show that new sequences from our construction are balanced or almost balanced and have optimal three level autocorrelation for the case when the polynomial z 1 sup d z sup d a can be transformed into the form z sup 2 c we also derive the distribution of autocorrelation values they take on the sequences satisfy constant on the coset property and we show that there are more than one characteristic phases with constant on the coset property some other interesting properties of those sequences are presented for the cases when the polynomial z 1 sup d z sup d a cannot be transformed into the form z sup 2 c we performed extensive computer search and results are summarized based on these results some open problems are formulated

utility maximizing resource control diffusion limit and asymptotic optimality for a two bottleneck model we study a stochastic network that consists of two servers shared by two classes of jobs class 1 jobs require a concurrent occupancy of both servers while class 2 jobs use only one server the traffic intensity is such that both servers are bottlenecks meaning the service capacity is equal to the offered workload the real time allocation of the service capacity among the job classes takes the form of a solution to an optimization problem that maximizes a utility function we derive the diffusion limit of the network and establish its asymptotic optimality in particular we identify a cost objective associated with the utility function and show that it is minimized at the diffusion limit by the utility maximizing allocation within a broad class of fair allocation schemes the model also highlights the key issues involved in multiple bottlenecks

trace representation of legendre sequences of mersenne prime period in this correspondence it is shown that legendre sequences of period p can be explicitly represented using the trace function defined on the finite field with 2 sup n elements whenever p 2 sup n 1 is prime for some n spl ges 3

binary pseudorandom sequences of period 2 sup m 1 with ideal autocorrelation generated by the polynomial z sup d z 1 sup d we present a construction for binary pseudorandom sequences of period 2 sup m 1 with ideal autocorrelation property using the polynomial z sup d z 1 sup d we show that the sequence obtained from the polynomial becomes an m sequence for certain values of d we also find a few values of d which yield new binary sequences with ideal autocorrelation property when m is 3k spl plusmn 1 where k is a positive integer these new sequences are represented using a trace function and the results are tabulated

the number of binary words avoiding abelian fourth powers grows exponentially we show that the number of binary words of length n avoiding abelian fourth powers grows faster than rn where r 21 16

new binary sequences with optimal autocorrelation magnitude new binary sequences of period n 4 2m 1 for even m ges 4 are found where the sequences are described by a 4 x 2m 1 array structure the new sequences are almost balanced and have four valued autocorrelation i e n 0 plusmn4 which is optimal with respect to autocorrelation magnitude the complete autocorrelation distribution and the exact linear complexity of the sequences are mathematically derived finally it is shown that the sequences are implemented by a combination of linear feedback shift registers and a simple logic

a stochastic network under proportional fair resource control diffusion limit with multiple bottlenecks we study a multiclass stochastic processing network operating under the so called proportional fair allocation scheme and following the head of the line processor sharing discipline specifically each server s capacity is shared among the job classes that require its service and it is allocated in every state of the network among the first waiting job of each class to maximize a log utility function we establish the limiting regime of the network under diffusion scaling allowing multiple bottlenecks in the network and relaxing some of the conditions required in prior studies we also identify the class of allocation schemes among which the proportional fair allocation minimizes a quadratic cost objective function of the diffusion scaled queue lengths and we illustrate the limitation of this asymptotic optimality through a counterexample

spectral methods for cross correlations of geometric sequences families of sequences with low pairwise shifted cross correlations are desirable for applications such as code division multiple access cdma communications often such sequences must have additional properties for specific applications several ad hoc constructions of such families exist in the literature but there are few systematic approaches to such sequence design we introduce a general method of constructing new families of sequences with bounded pairwise shifted cross correlations from old families of such sequences the bounds are obtained in terms of the maximum cross correlation in the old family and the walsh transform of certain functions

fluid limits for processor sharing queues with impatience we investigate a processor sharing queue with renewal arrivals and generally distributed service times impatient jobs may abandon the queue or renege before completing service the random time representing a job s patience has a general distribution and may be dependent on its initial service time requirement a scaling procedure that gives rise to a fluid model with nontrivial yet tractable steady state behavior is presented this fluid model captures many essential features of the underlying stochastic model and it is used to analyze the impact of impatience in processor sharing queues

generalization of gmw sequences and no sequences gmw sequences and families of no sequences are generalized generalized gmw sequences have ideal autocorrelation and balance properties and generalized no sequences also have optimal correlation properties in terms of welch s lower bound the linear spans of the generalized gmw and no sequences appear to be large although there is not at present a closed form expression for the linear span a count of the numbers of cyclically distinct generalized gmw sequences and generalized no sequences that can be constructed is provided generalized gmw sequences have also been found in the literature by klapper et al 1993 under the name cascaded gmw sequences

growth problems for avoidable words abstract a word w is said to avoid a word u if no block subword factor of w is the image of u under a homomorphism of free semigroups without unit the theory of words avoiding xx square free words has been much studied the word u is said to be avoidable on n letters if there are arbitrarily long words on an n letter alphabet that avoid u if u is avoidable on n letters for some n let u be the minimum possible such n we show that u has a linear bound in terms of the alphabet size of u we further show that there exists a word that is avoidable on four letters but not on three letters moreover if u is this word the number of words of length l on a u letter alphabet that avoid u has a polynomial bound in terms of l so that the question of the existence of such an example is resolved in the affirmative in contrast for xx the bound is known to be exponential

binary pseudorandom sequences of period 2 sup n 1 with ideal autocorrelation in this correspondence we present five new classes of binary sequences of period 2 sup n 1 with ideal autocorrelation these sequences which correspond to new cyclic hadamard difference sets were found by extensive computer search conjectures on the general construction of these sequences are formulated

fluid limits for bandwidth sharing networks in overload bandwidth sharing networks as considered by roberts and massoulie 28 roberts jw massoulie l 1998 bandwidth sharing and admission control for elastic traffic proc itc specialist seminar yokohama japan provide a natural modeling framework for describing the dynamic flow level interaction among elastic data transfers under mild assumptions it has been established that a wide family of so called fair bandwidth sharing strategies achieve stability in such networks provided that no individual link is overloaded in the present paper we focus on bandwidth sharing networks where the load on one or several of the links exceeds the capacity to characterize the overload behavior we examine the fluid limit which emerges when the flow dynamics are scaled in both space and time we derive a functional equation characterizing the fluid limit and show that any strictly positive solution must be unique which in particular implies the convergence of the scaled number of flows to the fluid limit for nonz

reflected brownian motion on an orthant we consider a k dimensional diffusion process z whose state space is the nonnegative orthant on the interior of the orthant z behaves like a kdimensional brownian motion with arbitrary covariance matrix and drift vector at each of the k 1 dimensional hyperplanes that form the boundary of the orthant z reflects instantaneously in a direction that is constant over that hyperplane there is no extant theory of multidimensional diffusion that applies to this process because the boundary of its state space is not smooth we adopt an approach that requires a restriction on the directions of reflection but reiman has shown that this restriction is met by all diffusions z arising as heavy traffic limits in open k station queuing networks our process z is defined as a path to path mapping of k dimensional brownian motion from this construction it follows that z is a continuous markov process and a semimartingale using the latter property we obtain a change of variable formula from which one can develop a complete analytical theory for the process z

a powerful abelian square free substitution over 4 letters in 1961 paul erdos posed the question whether abelian squares can be avoided in arbitrarily long words over a finite alphabet an abelian square is a non empty word uv where u and v are permutations anagrams of each other the case of the four letter alphabet s 4 a b c d turned out to be the most challenging and remained open until 1992 when the author presented an abelian square free a 2 free endomorphism g 8 5 of s 4 the size of this g 8 5 i e g 8 5 abcd is equal to 4x85 uniform modulus until recently all known methods for constructing arbitrarily long a 2 free words on s 4 have been based on the structure of g 8 5 and on the endomorphism g 9 8 of s 4 found in 2002 in this paper a great many new a 2 free endomorphisms of s 4 are reported the sizes of these endomorphisms range from 4x102 to 4x115 importantly twelve of the new a 2 free endomorphisms of modulus m 109 can be used to construct an a 2 free commutatively functional substitution s 1 0 9 of s 4 with 12 image words for each letter the properties of s 1 0 9 lead to a considerable improvement for the lower bound of the exponential growth of c n i e of the number of a 2 free words over 4 letters of length n it is obtained that c n b 5 email protected n with b 12 1 m 1 02306 originally in 1998 carpi established the exponential growth of c n by showing that c n b email protected n with b 2 1 9 t 2 1 9 8 5 3 8 5 1 000021 where t 85 3 85 is the modulus of the substitution that he constructs starting from g 8 5

new nonbinary sequences with ideal two level autocorrelation we find new families of nonbinary sequences of period p sup n 1 with symbols from a finite field f sub p for any prime p spl ges 3 the sequences have two level ideal autocorrelation and are generalizations of previously found ternary sequences with ideal autocorrelation difference sets with parameters p sup n 1 p 1 p sup n 1 1 p 1 p sup n 2 1 p 1 can also be derived from these sequences in a natural way

binary sequences with optimal autocorrelation sequences have important applications in ranging systems spread spectrum communication systems multi terminal system identification code division multiple access communication systems global positioning systems software testing circuit testing computer simulation and stream ciphers sequences and error correcting codes are also closely related in this paper we give a well rounded treatment of binary sequences with optimal autocorrelation we survey known ones and construct new ones

trace representation and linear complexity of binary e th power residue sequences of period p let p ef 1 be an odd prime for some e and e and let f be the finite field with fp elements in this paper we explicitly describe the trace representations of the binary characteristic sequences of period p of all the cyclic difference sets d which are some union of cosets of eth powers he in fp fp 0 for e 12 for this we define eth power residue sequences of period p which include all the binary characteristic sequences mentioned above as special cases and reduce the problem of determining their trace representations to that of determining the values of the generating polynomials of cosets of he in f at some primitive pth root of unity and some properties of these values are investigated based on these properties the trace representation and linear complexity not only of the characteristic sequences of all the known eth residue difference sets but of all the sixth power residue sequences are determined furthermore we have determined the linear complexity of a nonconstant eth power residue sequence for any e to be either p 1 or p whenever e p 1 n 1 where n is the order of 2 mod p

the proof of lin s conjecture via the decimation hadamard transform in 1998 lin presented a conjecture on a class of ternary sequences with ideal two level autocorrelation those sequences have a very simple structure i e their trace representation has two trace monomial terms in this paper we present a proof for the conjecture the mathematical tools employed are the second order multiplexing decimation hadamard transform stickelberger s theorem the teichmuller character and combinatorial techniques for enumerating the hamming weights of ternary numbers as a by product we also prove that the ternary sequences conjectured by lin are hadamard equivalent to ternary m sequences

the linear complexity of some binary sequences with three level autocorrelation binary sequences with good autocorrelation are needed in many applications a construction of binary sequences with three level autocorrelation was recently presented this construction is generic and powerful in the sense that many classes of binary sequences with three level autocorrelation could be obtained from any difference set with singer parameters the objective of this paper is to determine both the linear complexity and the minimal polynomial of two classes of binary sequences i e the class based on the singer difference set and the class based on the gmw difference set

cryptographic properties of the welch gong transformation sequence generators welch gong wg transformation sequences are binary sequences of period 2 sup n 1 with two level autocorrelation these sequences were discovered by golomb gong and gaal 1998 and they verified the validity of their construction for 5 spl les n spl les 20 later no chung and yun 1998 found another way to construct the wg sequences and verified their result for 5 spl les n spl les 20 dillon 1998 first proved this result for odd n and finally dobbertin and dillon 1999 proved it for even n in this paper we investigate a two faced property of the wg transformation sequences for application in stream ciphers and pseudorandom number generators one is to present the randomness or unpredictability of the wg transformation sequences the other is to exhibit the security properties of the wg transformations regarded as boolean functions in particular we prove that the wg transformation sequences in addition to the known two level autocorrelation and three level cross correlation with m sequences have the ideal 2 tuple distribution and large linear span increasing exponentially with n moreover it can be implemented efficiently this is the first type of pseudorandom sequences with good correlation statistic properties large linear span and efficient implementation when wg transformations are regarded as boolean functions they have high nonlinearity we derive a criterion for the boolean representation of wg transformations to be r resilient and show that they are at least 1 resilient under some basis of the finite field gf 2 sup n an algorithm to find such bases is given the degree and linear span of wg transformations are presented as well