Complementary Probability Research Articles

How linear reinforcement affects Donsker\u2019s theorem for empirical processes

A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability pin (0,1), {hat{U}}_{n+1} is sampled uniformly from {hat{U}}_1, ldots , {hat{U}}_n, and with complementary probability 1-p, {hat{U}}_{n+1} is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when p<1/2, and that a further rescaling is needed when p>1/2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.

A production inventory model with deteriorating items and retrial demands

This paper considers a single server perishable inventory system in which customers arrive in a homogeneous Poisson stream. The system has a production unit which produces a single item in an exponentially distributed time interval. At the time of arrival, a customer leads to service if the server is available with a positive level of inventory. Otherwise, the customer goes to a waiting place(orbit) of infinite capacity with pre-determined probability or exits the system with complementary probability. Each customer in the orbit tries to access the server in an exponentially distributed time interval. After every unsuccessful retrial, the customer returns to the orbit with a pre-allotted probability or is lost forever with complementary probability. An algorithmic solution to the problem is obtained using Matrix Analytic Method. The mean number of customer loss before and after entering the system, the rate of successful retrials among overall retrials and some other performance measures of the system are derived. The impacts of system parameters on different measures are numerically studied. A suitable cost function is constructed and the optimum control policy is numerically obtained.

Energy Harvesting in Unmanned Aerial Vehicle Networks With 3D Antenna Radiation Patterns

In this paper, an analytical framework is provided to analyze the energy coverage performance of unmanned aerial vehicle (UAV) energy harvesting networks with clustered user equipments (UEs). Locations of UEs are modeled as a Poisson Cluster Process (PCP), and UAVs are assumed to be located at a certain height above the center of user clusters. Hence, user-centric UAV deployments are addressed. Two different models are considered for the line-of-sight (LOS) probability function to compare their effects on the network performance. Moreover, antennas with doughnut-shaped radiation patterns are employed at both UAVs and UEs, and the impact of practical 3D antenna radiation patterns on the network performance is also investigated. Initially, the path loss of each tier is statistically described by deriving the complementary cumulative distribution function and probability density function. Following this, association probabilities with each tier are determined, and energy coverage probability of the UAV network is characterized in terms of key system and network parameters for UAV deployments both at a single height level and more generally at multiple heights. Through numerical results, we have shown that cluster size and UAV height play crucial roles on the energy coverage performance. Furthermore, energy coverage probability is significantly affected by the antenna orientation and number of UAVs in the network.

Fitting in and breaking up: A nonlinear version of coevolving voter models.

We investigate a nonlinear version of coevolving voter models, in which node states and network structure update as a coupled stochastic process. Most prior work on coevolving voter models has focused on linear update rules with fixed and homogeneous rewiring and adopting probabilities. By contrast, in our nonlinear version, the probability that a node rewires or adopts is a function of how well it "fits in" with the nodes in its neighborhood. To explore this idea, we incorporate a local-survey parameter σ_{i} that encodes the fraction of neighbors of an updating node i that share its opinion state. In an update, with probability σ_{i}^{q} (for some nonlinearity parameter q), the updating node rewires; with complementary probability 1-σ_{i}^{q}, the updating node adopts a new opinion state. We study this mechanism using three rewiring schemes: after an updating node deletes one of its discordant edges, it then either (1) "rewires-to-random" by choosing a new neighbor in a random process; (2) "rewires-to-same" by choosing a new neighbor in a random process from nodes that share its state; or (3) "rewires-to-none" by not rewiring at all (akin to "unfriending" on social media). We compare our nonlinear coevolving voter model to several existing linear coevolving voter models on various network architectures. Relative to those models, we find in our model that initial network topology plays a larger role in the dynamics and that the choice of rewiring mechanism plays a smaller role. A particularly interesting feature of our model is that, under certain conditions, the opinion state that is held initially by a minority of the nodes can effectively spread to almost every node in a network if the minority nodes view themselves as the majority. In light of this observation, we relate our results to recent work on the majority illusion in social networks.

A model for the competition between political mono-polarization and bi-polarization.

We investigate the phenomena of political bi-polarization in a population of interacting agents by means of a generalized version of the model introduced by Vazquez et al. [Phys. Rev. E 101, 012101 (2020)] for the dynamics of voting intention. Each agent has a propensity p in [0,1] to vote for one of two political candidates. In an iteration step, two randomly chosen agents i and j with respective propensities pi and pj interact, and then pi either increases by an amount h>0 with a probability that is a nonlinear function of pi and pj or decreases by h with the complementary probability. We assume that each agent can interact with any other agent (all-to-all interactions). We study the behavior of the system under variations of a parameter q≥0 that measures the nonlinearity of the propensity update rule. We focus on the stability properties of the two distinct stationary states: mono-polarization in which all agents share the same extreme propensity (0 or 1), and bi-polarization where the population is divided into two groups with opposite and extreme propensities. We find that the bi-polarized state is stable for q<qc, while the mono-polarized state is stable for q>qc, where qc(h) is a transition value that decreases as h decreases. We develop a rate equation approach whose stability analysis reveals that qc vanishes when h becomes infinitesimally small. This result is supported by the analysis of a transport equation derived in the continuum h→0 limit. We also show by Monte Carlo simulations that the mean time τ to reach mono-polarization in a system of size N scales as τ∼Nα at qc , where α is a nonuniversal exponent that depends on h.

Symmetrical threshold model with independence on random graphs.

We study the homogeneous symmetrical threshold model with independence (noise) by pair approximation and Monte Carlo simulations on Erdős-Rényi and Watts-Strogatz graphs. The model is a modified version of the famous Granovetter's threshold model: with probability p a voter acts independently, i.e., takes randomly one of two states ±1; with complementary probability 1-p, a voter takes a given state, if a sufficiently large fraction (above a given threshold r) of individuals in its neighborhood is in this state. We show that the character of the phase transition, induced by the noise parameter p, depends on the threshold r, as well as graph's parameters. For r=0.5 only continuous phase transitions are observed, whereas for r>0.5 discontinuous phase transitions also are possible. The hysteresis increases with the average degree 〈k〉 and the rewriting parameter β. On the other hand, the dependence between the width of the hysteresis and the threshold r is nonmonotonic. The value of r, for which the maximum hysteresis is observed, overlaps pretty well with the size of the majority used for the descriptive norms in order to manipulate people within social experiments. We put the results obtained within this paper into a broader picture and discuss them in the context of two other models of binary opinions: the majority-vote and the q-voter model. Finally, we discuss why the appearance of social hysteresis in models of opinion dynamics is desirable.

A Multiserver Tandem Queue with a Specialist Server Operating with a Vacation Strategy

In the considered queueing system, the service is provided at two stations, station 1 and station 2 operating in tandem. Station 1 is a multi-server station with c identical servers working in parallel, and station 2 is equipped with a single server called the specialist server. The service times of each of the servers at station 1 follow an exponential distribution. The specialist server has phase-type distributed service times. Customers arrive at station 1 according to a Markovian arrival process. An arriving customer enters directly into service at station 1 if at least one of the servers is idle, otherwise joins an infinite queue. After receiving service at station 1, customers either proceed to station 2 or can exit the system. There is a finite buffer between two stations. When the buffer is not full, a customer coming out of the station 1 joins the buffer with a probability p or leaves the system with the complementary probability 1 − p. If the buffer is full, then all the customers coming out of station 1 are lost forever. The server at the station 2 will be turned on only if the number of customers in the buffer reaches a threshold. Once the server is turned on, the service will be rendered until the buffer is emptied. A stability condition for this system is established and the stationary distribution is obtained using matrix analytic methods. Various performance measures are also calculated. Our model is motivated by a hospital situation where station 1 represents the causalty clinic and the specialist server represents an expert giving a consultation upon the request of a threshold number of patients.

Acoustic emissions in compression of building materials: q-statistics enables the anticipation of the breakdown point

In this paper we present experimental results concerning Acoustic Emission (AE) recorded during cyclic compression tests on two different kinds of brittle building materials, namely concrete and basalt. The AE inter-event times were investigated through a non-extensive statistical mechanics analysis which shows that their complementary cumulative probability distributions follow q -exponential laws. The entropic index q and the relaxation parameter β q ≡ 1∕T q , obtained by fitting the experimental data, exhibit systematic changes during the various stages of the failure process, namely (q , T q ) linearly align. The T q = 0 point corresponds to the macroscopic breakdown of the material. The slope, including its sign, of the linear alignment appears to depend on the chemical and mechanical properties of the sample. These results provide an insight on the warning signs of the incipient failure of building materials and could therefore be used in monitoring the health of existing structures such as buildings and bridges.

Survival probabilities in biased random walks: To restart or not to restart? That is the question

The time-dependent survival probability function S(t;x0,q) of biased Sisyphus random walkers, who at each time step have a finite probability q to step towards an absorbing trap at the origin and a complementary probability 1−q to return to their initial position x0, is derived analytically. In particular, we explicitly prove that the survival probability function of the walkers decays exponentially at asymptotically late times. Interestingly, our analysis reveals the fact that, for a given value q of the biased jumping probability, the survival probability function S(t;x0,q) is characterized by a critical (marginal) value x0crit(q) of the initial gap between the walkers and the trap, above which the late-time survival probability of the biased Sisyphus random walkers is larger than the corresponding survival probability of standard random walkers.

Evidential reasoning based on imitation and aspiration information in strategy learning promotes cooperation in optional spatial public goods game

Abstract Mechanisms of strategy learning have significant impact on the equilibrium state of the system in evolutionary games. As two common individual strategy updating rules, pairwise imitation and aspiration-driven updating have been widely used in various models of evolutionary games. In previous studies, it is generally assumed that individuals only adopt a single rule to update their strategies. Recently, there have been researches on the combination of the two rules, such as choosing one rule with a certain probability and choosing the other with its complementary probability. Under this combination rule, however, each individual still utilizes single-source information for strategy selection each time. In this paper, we extend the combination of the two rules from the perspective of multi-source information fusion and also consider the ignorance situation that may be contained in the obtained information. Since evidence theory has the advantage in representing knowledge with uncertainty and unknown, we introduce the tool of evidential reasoning to construct a new strategy learning rule where information from imitation and aspiration-driven updating is taken as two pieces of evidence, and individuals update strategies via evidence reasoning. The effect of the evidential-reasoning-based new rule is investigated in detail in the spatial public goods game with optional participation, where individuals have three alternative strategies, namely, cooperation, defection, and loner strategy. Through numerical simulations, we find that evidential reasoning can effectively improve the performance of both imitation and aspiration-driven updating in promoting the emergence of cooperation within a wide range of fusion weights. Particularly, in the region of low-synergy factors, cooperation can also be enhanced. These results can complement our understanding of the emergence of cooperation from the perspective of individual learning.

Analysis of a Single Server Retrial Queue with Server Vacation and Two Waiting Buffers Based on ATM Networks

This paper presents a discrete-time Geo/G/1 retrial queue with two waiting buffers to model an ATM network, in which the server begins a single vacation in cases where the system is empty at the instant of a service completion. New arriving customer who finds the server being on vacation can decide to either enter the retrial buffer with some probability p or leave the system with complementary probability 1−p. But the new arriving customer can begin its service immediately if he finds the server idle and join the original buffer if he finds the server busy. We first carry out an extensive analysis of the model by using the supplementary variable method and the generating function approach, and give some performance measures, such as server’s state probabilities and mean queue lengths in the original buffer, retrial buffer, and in the system. Secondly, we give the generating function of the sojourn time of a customer in the system and prove that Little’s law still holds in our model. Sensitivity analysis and cost optimization are finally given for illustrative purposes.

Analysis of Queueing System MMPP/M/K/K with Delayed Feedback

The model of multi-channel queuing system with Markov modulated Poisson process (MMPP) flow and delayed feedback is considered. After the customer is served completely, they will decide either to join the retrial group again for another service (feedback) with some state-dependent probability or to leave the system forever with complimentary probability. Feedback calls organize an orbit of repeated calls (r-calls). If upon arrival of an r-call all the channels of the system are busy, then it either leaves the system with some state-dependent probability or with a complementary probability returns to orbit. Methods to calculate the steady-state probabilities of the appropriate three-dimensional Markov chain as well as performance measures of investigated system are developed. Results of numerical experiments are demonstrated.

Availability Modelling of Cluster-Based System with Software Aging and Optional Rejuvenation Policy

Abstract This investigation deals with modeling and availability analysis of cluster-based system inflicted with software aging. Software aging is a phenomenon in which a software system shows performance degradation with time and finally results in software failures. To cope up with this phenomenon, rejuvenation is an innovative concept to recover from software failures. As failures occur, server has the option either to take essential rejuvenation with probability p or may opt for optional rejuvenation with complementary probability q. To achieve high availability of the system, the concept of clustering is also taken into consideration. In this study, restart, reboot and standby concept is used for reducing the downtime cost. The sensitivity analysis of different parameters on system availability has been examined numerically. By integrating clustering, software aging and rejuvenation, the researchers intended to increase the availability and decrease the down time.

Bilateral bargaining with endogenous status quo

We consider a non-cooperative bargaining game where in each round, if a proposal is rejected, with a probability, the allocation implemented in the previous round is implemented as a status quo and the game proceeds to the next round, and with the complementary probability, no allocations are implemented and the negotiations break down. We show that there exists a unique stationary subgame perfect equilibrium such that in any round, a proposer offers a proposal such that the responder is indifferent between accepting and rejecting it and an agreement is achieved. If the probability of breakdown is close to zero, the sequence of equilibrium allocations evolves from the initial status quo to an allocation near the equilibrium allocation of the Rubinstein bargaining game over time. The effect of the initial status quo on the equilibrium allocations vanishes over time.

Stochastic modeling of quality of systems operating in a heterogeneous environment

AbstractWe consider systems that are subject to an external mixed Poisson shock process. Each shock can result in a failure of a system with a given probability and is survived with the complementary probability. Each shock additionally decreases the quality function that describes the performance of a system, thus forming the corresponding stochastic process. Expectations (unconditional and conditional on survival) and relevant variability characteristics for the stochastic quality function are derived. Some monotonicity properties of the conditional quality function are investigated and the future values of this function are derived.

Rare Events Queueing System – REQS

The paper deals with the queueing system for customers with Poisson’s input current intensity l and two service modes: in the regular service regime of intensity control m, customers are served with probability p»1 and in the special regime of servicing the special customers with intensity x. The special customers access the REQS with complementary probability (1-p)»0. The special customer service is analogous to a rare event. The standard methodology has developed analytical patterns for the stationary of REQS with one service channel and an infinite number of positions in the queue. The analysis of the work of REQS indicates that it is for favorable metering parameters r=l/m>2, queueing system is resistant to collapse when a occurrence occurs. However, the regular time losses of the regular customers in the REQS are extremely high. For this reason, it is the first time that the period of stabilization of the system is promoted which represents the time interval service the completion of the special customers until the REQS. The analytical apparatus of the queueing system has shown excellent adaptability to the heterogeneous demands of services m and special customers with low service intensity x, where m>x. The system can be applied to checkpoint calculations, the traffic cuts due to accidents, incidents to industrial systems, ie, the rare events due to anthropogenic and technical factors in intervals of 10-4 do 10-6. The model is not intended for natural hazards.

A Version of Herbert A. Simon’s Model with Slowly Fading Memory and Its Connections to Branching Processes

Construct recursively a long string of words w1. .. wn, such that at each step k, w k+1 is a new word with a fixed probability p $\in$ (0, 1), and repeats some preceding word with complementary probability 1 -- p. More precisely, given a repetition occurs, w k+1 repeats the j-th word with probability proportional to j $\alpha$ for j = 1,. .. , k. We show that the proportion of distinct words occurring exactly times converges as the length n of the string goes to infinity to some probability mass function in the variable $\ge$ 1, whose tail decays as a power function when 1 -- p > $\alpha$/(1 + $\alpha$), and exponentially fast when 1 -- p < $\alpha$/(1 + $\alpha$).

A central limit theorem for a Markov-modulated infinite-server queue with batch Poisson arrivals and binomial catastrophes

This paper considers the stationary queue length distribution of a Markov-modulated MX/M/∞queue with binomial catastrophes. When a binomial catastrophe occurs, each customer is either removed with a probability or is retained with the complementary probability. We focus on our model under a heavy traffic regime because its exact analysis is difficult if not impossible. We establish a central limit theorem for the stationary queue length of our model in a heavy traffic regime. The central limit theorem can be used to approximate the queue length distribution of our model with large arrival rates.

Diffusion-Based Molecular Communication Channel in Presence of a Probabilistic Absorber: Single Receptor Model and Congestion Analysis.

In this paper, a diffusion-based molecular communication channel is modeled in presence of a probabilistic absorber. The probabilistic absorber is an absorber which absorbs molecules upon collision with probability q . With random walk analysis, the discrete probability function of particle location in the presence of a probabilistic absorber can be found. Then, a continuous probability function is fitted to this Markov-based results with introducing several fitting parameters to the known probability function of particle location in an unbounded environment without an absorbing barrier. With this approach, a single receptor is modeled as an M/M/1/1 queue in which q represents the complementary blocking probability and the mean service time is the mean trafficking time. Therefore, we are able to model the stochastic nature of ligand-receptor binding, which comes from the incapability of a receptor to receive all molecules in its space; and also known as receptor occupancy. The proper consideration of the absorption effect leads to the accurate calculation of the concentration at the desired site, which is definitely less than the concentration obtained when neglecting it. These findings can have a crucial role in designing drug delivery systems in which determining the optimal rate of the drug transmitting nanomachines is critical to avoid toxicity while maintaining effectiveness.

Majority vote model with ancillary noise in complex networks

We analyze the properties of the majority-vote (MV) model with an additional noise in which a local spin can be changed independently of its neighborhood. In the standard MV, one of the simplest nonequilibrium systems exhibiting an order–disorder phase transition, spins are aligned with their local majority with probability 1−f, and with complementary probability f, the majority rule is not followed. In the noisy MV (NMV), a random spin flip is succeeded with probability p (with complementary 1−p the usual MV rule is accomplished). Such extra ingredient was considered by Vieira and Crokidakis (2016) for the square lattice. Here, we generalize the NMV for arbitrary networks, including homogeneous [random regular (RR) and Erdös–Renyi (ER)] and heterogeneous [Barabasi–Albert (BA)] structures, through mean-field calculations and numerical simulations. Results coming from both approaches are in excellent agreement with each other, revealing that the presence of additional noise does not affect the classification of phase transition, which remains continuous irrespective of the network degree and its distribution. The critical point and the threshold probability pt marking the disappearance of the ordered phase depend on the node distribution and increase with the connectivity k. The critical behavior, investigated numerically, exhibits a common set of critical exponents for RR and ER topologies, but different from BA and regular lattices. Finally, our results indicate that (in contrary to a previous proposition) there is no first-order transition in the NMV for large k.