the asymptotic optimality of the lpt rule for the problem of minimizing makespan on parallel machines of different speed the behaviour of list scheduling rules is subjected to a probabilistic analysis under the assumption that the processing requirements of the jobs are independent identically distributed nonnegative random variables under mild conditions on the probability distribution we obtain strong asymptotic optimality results for the lpt longest processing time rule in which the jobs are assigned to the machines in order of nonincreasing processing requirements

stochastic analysis of a modified first fit decreasing packing we make a stochastic analysis of a modified version mffd of first fit decreasing in which each bin is closed after it receives its first fallback item consider a probability measure on 0 1 and independent random variables x1 xn distributed according to let rn rx1 xn be the number of unit size bins that mffd needs to pack items of size x1 xn we prove that c limn ern n exists and that the random variable rn nc n converges in distribution the main tools are deterministic inequalities concerning mffd that might be of independent interest

using dual approximation algorithms for scheduling problems theoretical and practical results the problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well studied problem in the theory of approximation algorithms for np hard optimization problems in this paper the strongest possible type of result for this problem a polynomial approximation scheme is presented more precisely for each e an algorithm that runs in time o n e 1 e 2 and has relative error at most e is given in addition more practical algorithms for e 1 5 2 k and e 1 6 2 k which have running times o n k log n and o n km 4 log n are presented the techniques of analysis used in proving these results are extremely simple especially in comparison with the baroque weighting techniques used previously the scheme is based on a new approach to constructing approximation algorithms which is called dual approximation algorithms where the aim is to find superoptimal but infeasible solutions and the performance is measured by the degree of infeasibility allowed this notion should find wide applicability in its own right and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive

asymptotic behavior of solutions to a hyperbolic elliptic coupled system in multi dimensional radiating gas this paper is concerned with the asymptotic behavior of solutions to the cauchy problem of a hyperbolic elliptic coupled system in the multi dimensional radiating gas ut a u2 divq 0 divq q u 0 with initial data u x1 xn 0 u0 x1 xn u x1 first for the case with the same end states u u 0 we prove the existence and uniqueness of the global solutions to the above cauchy problem by combining some a priori estimates and the local existence based on the continuity argument then lp convergence rates of solutions are respectively obtained by applying l2 energy method for n 1 2 3 and lp energy method for 3 n 8 and interpolation inequality furthermore by semigroup argument we obtain the decay rates to the diffusion waves for 1 n 8 secondly for the case with the different end states u u our main concern is that the corresponding cauchy problem in n dimensional space n 1 2 3 behaviors like planar rarefaction waves its convergence rate is also obtained by l2 energy method and l1 estimate

time fractional telegraph equations and telegraph processes with brownian time we study the fundamental solutions to time fractional telegraph equations of order 2 we are able to obtain the fourier transform of the solutions for any and to give a representation of their inverse in terms of stable densities for the special case 1 2 we can show that the fundamental solution is the distribution of a telegraph process with brownian time in a special case this becomes the density of the iterated brownian motion which is therefore the fundamental solution to a fractional diffusion equation of order 1 2 with respect to time

optimal bin packing with items of random sizes consider a probability on 0 1 and i i d random variables x1 x2 xn distributed like let qn denote the optimal minimum number of unit size bins needed to pack items of size x1 x2 xn we characterize the class of which have the property that limn qn n e x1 a s or equivalently that the expected level of occupancy of bins converges to one

probabilistic analysis of a generalized bin packing problem and applications we give a unified probabilistic analysis for a general class of bin packing problems by directly analyzing corresponding mathematical programs in this general class of packing problems objects are described by a given number of attribute values some attributes may be discrete others may be continuous bins are sets of objects and the collection of feasible bins is merely required to satisfy some general consistency properties we characterize the asymptotic optimal value as the value of an easily specified linear program whose size is independent of the number of objects to be packed or as the limit of a sequence of such linear program values we also provide bounds for the rate of convergence of the average cost to its asymptotic value the analysis suggests an a s asymptotically optimal heuristic that runs in linear time the heuristics can be designed to be asymptotically optimal while still running in polynomial time we also show that in several important cases the algorithm has both po

multidimensional optimal bin packing with items of random size consider a probability measure on 0 1 d and independent random variables x 1 x n distributed according to let vn vnx 1 x n resp rn rnx 1 x n be the minimal number of unit size d dimensional bins needed to pack items of size x 1 x n in the vector packing resp rectangular packing problem for a large class of distributions we characterize v limn vn n and r limn rn n

tight bounds and probabilistic analysis of two heuristics for parallel processor scheduling we study the partitioning problem consisting in partitioning a sublist of n positive numbers into m disjoint sublists such that the maximum sublist is minimized this is equivalent to minimizing the completion time of n jobs on m parallel identical processors we establish upper bounds on the deviation from optimality of two heuristics the well known lpt heuristic and the on line rlp heuristic these bounds serve to establish a probabilistic analysis of these heuristics for both of them the absolute deviation from optimality remains finite when the size of the list of numbers becomes infinite this is a stronger result than previous convergence theorems and it is valid whenever the processing times are iid random variables with finite mean and arbitrary distributions

fractional non linear linear and sublinear death processes this paper is devoted to the study of a fractional version of non linear open image in new window t 0 linear m t t 0 and sublinear mathfrak m nu t t 0 death processes fractionality is introduced by replacing the usual integer order derivative in the difference differential equations governing the state probabilities with the fractional derivative understood in the sense of dzhrbashyan caputo we derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values a useful subordination relation is also proved allowing us to express the death processes as compositions of their classical counterparts with the random time process t2 t t 0 this random time has one dimensional distribution which is the folded solution to a cauchy problem of the fractional diffusion equation

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

population processes sampled at random times in this paper we study the iterated birth process of which we examine the first passage time distributions and the hitting probabilities furthermore linear birth processes linear and sublinear death processes at poisson times are investigated in particular we study the hitting times in all cases and examine their long range behavior the time changed population models considered here display upward birth process and downward jumps death processes of arbitrary size and for this reason can be adopted as adequate models in ecology epidemics and finance situations under stress conditions

probabilistic analysis of bin packing heuristics in this paper we present both a probabilistic and statistical analysis of several bin packing heuristics with mild conditions on the distribution of problem data we show that the ratio of the expected number of bins packed for these heuristics to the number of elements converges to some constant this constant can either be derived analytically or estimated through simulation we analyze the variance in performance of these heuristics and we further show that as the number of elements n increases the probability that the number of bins required for these heuristics exceeds any fixed percentage of their expected performance goes to zero

stochastic solution of space time fractional diffusion equations classical and anomalous diffusion equations employ integer derivatives fractional derivatives and other pseudodifferential operators in space in this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are mittag leffler type this leads to explicit solutions for space time fractional diffusion equations with multiscaling space fractional derivatives and additional insight into the meaning of these equations

a general lower bound for the makespan problem abstract the problem of scheduling jobs on identical parallel processors to minimize makespan is np hard we introduce a general lower bound which includes as special cases bounds that either dominate or are equivalent to lower bounds previously proposed in the literature the general lower bound unifies earlier results and can be used to identify new bounds improve previously proposed bounds and extend bounds for the problem with identical processor ready times to the problem with variable processor ready times

lower bounds and heuristic algorithms for the ki partitioning problem we consider the problem of partitioning a set of positive integers values into a given number of subsets each having an associated cardinality limit so that the maximum sum of values in a subset is minimized and the number of values in each subset does not exceed the corresponding limit the problem is related to scheduling and bin packing problems we give combinatorial lower bounds reduction criteria constructive heuristics a scatter search approach and a lower bound based on column generation the outcome of extensive computational experiments is presented

an analysis of the lpt algorithm for the max min and the min ratio partition problems given a set of positive numbers the max min partition problem asks for a k partition such that the minimum part is maximized the min ratio partition problem has the similar definition but the objective is to minimize the ratio of the maximum to the minimum parts in this paper we analyze the performances of the longest processing time lpt algorithm for the two problems we show that the tight bounds of the lpt are respectively 4k 2 3k 1 and 7 5

on a fractional binomial process the classical binomial process has been studied by jakeman j phys a 23 2815 2825 1990 and the references therein and has been used to characterize a series of radiation states in quantum optics in particular he studied a classical birth death process where the chance of birth is proportional to the difference between a larger fixed number and the number of individuals present it is shown that at large times an equilibrium is reached which follows a binomial process in this paper the classical binomial process is generalized using the techniques of fractional calculus and is called the fractional binomial process the fractional binomial process is shown to preserve the binomial limit at large times while expanding the class of models that include non binomial fluctuations non markovian at regular and small times as a direct consequence the generality of the fractional binomial model makes the proposed model more desirable than its classical counterpart in describing real physical processes more statistical properties are also derived

turnpike optimality of smith s rule in parallel machines stochastic scheduling consider scheduling a batch of jobs with stochastic processing times on parallel machines with minimization of expected weighted flowtime as objective smith s rule which orders job starts by decreasing ratio of weight to expected processing time provides a natural heuristic for this problem in a previous paper we have found an upper bound for the difference between the expected objective under smith s rule and under the optimal strategy in this paper we find an upper bound on the expected number of times that smith s rule differs from the optimal decision under some mild and reasonable assumptions these bounds remain constant when the number of jobs increases hence under these conditions smith s rule is asymptotically optimal and it has a turnpike optimality property that is smith s rule is optimal most of the time

minmax scheduling with job classes and earliness tardiness costs abstract we address scheduling problems with job dependent due dates and general possibly nonlinear and asymmetric earliness and tardiness costs the number of distinct due dates is substantially smaller than the number of jobs thus jobs are partitioned to classes where all jobs of a given class share a common due date we consider the settings of a single machine and parallel identical machines our objective is of a minmax type i e we seek a schedule that minimizes the maximum earliness tardiness cost among all jobs we introduce a nonlinear program that needs to be solved in order to obtain an optimal solution for the single machine case the case of parallel identical machines is np hard even for two machines linear costs and a single common due date we introduce an lpt based largest processing time first heuristic and a simple lower bound on the optimal cost both the heuristic and the lower bound are asymptotically accurate under very general conditions our numerical tests indicate that the heuristic produces very close to optimal schedules in all settings

some distributions that allow perfect packing a probability distribution on 0 1 allows perfect packing if n items of size x 1 x n independent and identically distributed according to can be packed in unit size bins in such a way that the expected wasted space is o n a large class of distributions that allow perfect packing is exhibited as a corollary the intervals a b for which the uniform distribution on a b allows perfect packing are determined

fractional diffusion equations and processes with randomly varying time in this paper the solutions u u x t to fractional diffusion equations of order 0 v 2 are analyzed and interpreted as densities of the composition of various types of stochastic processes for the fractional equations of order v 1 2 n n 1 we show that the solutions u 1 2 n correspond to the distribution of the n times iterated brownian motion for these processes the distributions of the maximum and of the sojourn time are explicitly given the case of fractional equations of order v 2 3 n n 1 is also investigated and related to brownian motion and processes with densities expressed in terms of airy functions in the general case we show that u coincides with the distribution of brownian motion with random time or of different processes with a brownian time the interplay between the solutions u and stable distributions is also explored interesting cases involving the bilateral exponential distribution are obtained in the limit

a note on expected makespans for largest first sequences of independent tasks on two processors we consider the scheduling of sets of n simultaneously available tasks on two identical processors the task execution times are assumed to be independent samples from the uniform distribution on 0 1 we analyze the expected makespan schedule length performance of two well known largest task first nonpreemptive approximation rules with the largest first lf rule tasks are assigned in nonincreasing order of execution time to the processors as they become available with the restricted largest first rlf rule tasks are assigned in pairs one to a processor the larger task of a pair is the first to be scheduled if n is odd the last task is simply assigned to the earner finishing processor in the schedule for the first n 1 tasks we prove that the expected makespan for lf is bounded by frac n 4 frac e 2n 1 and for rlf it is precisely frac n 4 frac 1 2n 1 from these results simple bounds are derived on the ratio of the expected performance of lf and rlf to the expected performance of an optimization rule finally a lower bound on expected lf makespans is shown to be frac n 4 frac 1 4n 1

non uniqueness of solutions to the cauchy problem for semilinear heat equations with singular initial data the cauchy problem for semilinear heat equations with singular initial data is studied where n 2 0 is a parameter and a 0 a 0 we show that when p n 2 n and n 2 p n 2 there exists a positive constant such that the problem has two positive self similar solutions and with if and no positive self similar solutions if furthermore for each fixed and in l rn as 0 where w0 is a non unique solution to the problem with zero initial data which is constructed by haraux and weissler

a note on fractional linear pure birth and pure death processes in epidemic models abstract in this note we highlight the role of fractional linear birth and linear death processes recently studied in orsingher et al 2010 5 and orsingher and polito 2010 6 in relation to epidemic models with empirical power law distribution of the events taking inspiration from a formal analogy between the equation for self consistency of the epidemic type aftershock sequences etas model and the fractional differential equation describing the mean value of fractional linear growth processes we show some interesting applications of fractional modelling in studying ab initio epidemic processes without the assumption of any empirical distribution we also show that in the framework of fractional modelling subcritical regimes can be linked to linear fractional death processes and supercritical regimes to linear fractional birth processes moreover we discuss a simple toy model in order to underline the possible application of these stochastic growth models to more general epidemic phenomena such as tumoral growth

average case analysis of cutting and packing in two dimensions abstract stock cutting and packing in two dimensions is a source of problems of great practical significance much has been written on the subject including several surveys but the focus of the mathematically oriented research has been largely on combinatorial optimization and worst case analysis recently substantial progress has been made in the average case analysis of algorithms for two dimensional packing this paper reviews selected illustrative results in this area of research and discusses desirable extensions and open problems several cutting and packing algorithms are defined and the results of their probabilistic analysis reported from the presentation of this material it is clear that the field is still in early stages of development the algorithms and probability models tend to be simplistic and stimates of performance are far more common that exact measures in spite of these limitations valuable and in some cases unexpected insights have emerged

the differencing algorithm ldm for partitioning a proof of a conjecture of karmarkar and karp the algorithm ldm largest differencing method divides a list of n random items into two blocks the parameter of interest is the expected difference between the two block sums it is shown that if the items are i i d and uniform then the rate of convergence of this parameter to zero is n log n an algorithm for balanced partitioning is constructed with the same rate of convergence to zero

martingale inequalities and np complete problems we use martingale inequalities in the probabilistic analysis of stochastic models of np complete problems we give examples of applications to the bin packing and traveling salesman problems in both cases we obtain sharp bounds on pr un eun t where un is the objective function value of the problem with n random data

coupled continuous time random walks in finance continuous time random walks ctrws are used in physics to model anomalous diffusion by incorporating a random waiting time between particle jumps in finance the particle jumps are log returns and the waiting times measure delay between transactions these two random variables log return and waiting time are typically not independent for these coupled ctrw models we can now compute the limiting stochastic process just like brownian motion is the limit of a simple random walk even in the case of heavy tailed power law price jumps and or waiting times the probability density functions for this limit process solve fractional partial differential equations in some cases these equations can be explicitly solved to yield descriptions of long term price changes based on a high resolution model of individual trades that includes the statistical dependence between waiting times and the subsequent log returns in the heavy tailed case this involves operator stable space time random vectors that generalize the familiar stable models in this paper we will review the fundamental theory and present two applications with tick by tick stock and futures data

blow up and global existence of solutions for a model system of the radiating gas abstract in this paper we are concerned with the cauchy problem for a model system of the radiating gas in multi dimensional space we first show the blow up of classical solutions to the nonlinear cauchy problem by employing the characteristic method on the basis of this blow up result when the initial data u 0 x satisfies the smallness conditions on u 0 h s but not on u 0 l 2 the global existence of solutions to the cauchy problem is established our approach is based on a low frequency high frequency decomposition and the classical energy method furthermore by taking a time frequency decomposition we prove the optimal time decay rates of the solutions

average performance of greedy heuristics for the integer knapsack problem abstract this paper derives a lower bound on the average performance of a total value greedy heuristic for the integer knapsack problem this heuristic selects items in order of their maximum possible contribution to the solution value at each stage we show that as for the worst case bound the average performance bound for the total value heuristic dominates the corresponding bound for the density ordered greedy heuristic

decay rates to the planar rarefaction waves for a model system of the radiating gas in n dimensions abstract in this paper we study the asymptotic decay rates to the planar rarefaction waves to the cauchy problem for a hyperbolic elliptic coupled system called as a model system of the radiating gas in r n n 3 4 5 if the initial perturbations corresponding to the planar rarefaction waves are sufficiently small in h 2 l 1 w 2 6 r n the analysis is based on the l p energy method and several special interpolation inequalities

asymptotic methods in the probabilistic analysis of sequencing and packing heuristics recently there has been considerable interest in the average case performance of heuristics this paper pursues that interest where it concerns sequencing and packing problems in particular we survey the methods that have been used to obtain formal probabilistic analyses of heuristics for makespan scheduling and one dimensional bin packing in so doing we present many of the key results in these research areas

on generalized fractional kinetic equations abstract in a recent paper saxena et al astro phys space sci 282 2002 281 developed solutions of generalized fractional kinetic equations in terms of mittag leffler functions the object of the present paper is to derive the solution of further generalized fractional kinetic equations their relation to fundamental laws of physics is briefly discussed results are obtained in a compact form in terms of generalized mittag leffler functions and a number of representations of these functions which are widely distributed in the literature are compiled for the first time

optimal bin packing with items of random sizes iii consider a probability measure mu on 0 1 and independent random variables x 1 cdots x n distributed according to mu let q n q n x 1 cdots x n be the minimum number of unit size bins needed to pack items of size x 1 cdots x n let c mu lim n to infty e q n n in this paper it is proved that the random variable q n nc mu sqrt n converges in distribution the limit is identified as a distribution of the supremum of a certain gaussian process canonically attached to mu

average case performance analysis of an approximation algorithm for maximum subset sum using recurrence relations given a positive integer m and a set s x1 x2 xn of positive integers the maximum subset sum problem is to find a subset s of s such that x s x is maximized under the constraint that the summation is no larger than m in addition the cardinality of s is also a maximum among all subsets of s which achieve the maximum subset sum this problem is known to be np hard we analyze the average case performance of a simple on line approximation algorithm assuming that all integers in s are independent and have the same probability distribution we develop a general methodology i e using recurrence relations to evaluate the expected values of the content and the cardinality of s for any distribution the maximum subset sum problem has important applications especially in static job scheduling in multiprogrammed parallel systems the algorithm studied can also be easily adapted for dynamic job scheduling in such systems

random walk and the area below its path a simple random walk with reflected origin is considered the walk starts at the origin and it must return to the origin at time 2n we show that the expected area below the path of this walk is n22n 1 cn2n if however the walk is required to return to the origin for the first time at time 2n then the expected area below the path of this bernoulli excursion is 2n 1 22n 1 cn2n we also show that if v1 vn is the order statistics based on a sample of size n from a uniform distribution over 0 1 and that if u1 un is another independent set of order statistics from the same distribution then e i 1n vi ui n22n 1 2n 1 cn2n we use this result to find an average case performance of the earliest due date edd heuristic for one machine scheduling problem with earliness and tardiness penalties we also apply some of the results to larson s queue inference engine 1990

positive solutions for semilinear elliptic equations with singular forcing terms abstract we consider the existence of solutions to the semilinear elliptic problem u u u p i 1 m c i a i in d r n u 0 a e in r n and u x 0 as x with prescribed given finite points a i i 1 m in r n and positive numbers c i i 1 m where n 3 1 p n n 2 0 is a parameter and a is the dirac delta function supported at a r n we reduce the problem to the problem in h 1 r n c 0 r n in terms of auxiliary functions and then show the existence of a positive constant 0 such that has at least two solutions if 0 a unique solution if and no solution if