Demetres D. Kouvatsos

A maximum entropy analysis of the M/G/1 and G/M/1 queueing systems at equilibrium

SummaryThe M/G/1 and G/M/1 queueing models are of great interest in the performance analysis of computer systems. For both models, the equilibrium solution for the number of jobs in the system varies with the probability distribution function representing the general (G-type) distribution. Even in the presence of empirical data, the characterisation of this function involves a degree of arbitrariness that may cause some variation in the performance metrics.In this paper maximum entropy formalism is used to analyse the M/G/1- and G/M/1-queueing systems at equilibrium. A unique product form solution for the number of jobs in the M/G/1 system is derived and the corresponding service time distribution is determined. This solution is also presented as a limit of a sequence of maximum entropy solutions to two-stage M/G/1 systems. Furthermore, the maximum entropy solution to the G/M/1 queueing system is established and favourable comparisons with the method of stages and the diffusion approximation are made. It is also shown that the maximum entropy M/G/1- and G/M/1-systems satisfy local balance. Comments on the results so far obtained and their implications to the analysis of general queueing systems are included.

Maximum entropy and the G / G /1/ N queue

SummaryA new “hybrid” analytic framework, based on the principle of maximum entropy, is used to derive a closed form expression for the queue length distribution of a G/G/1 finite capacity queue. It is shown that “Birth-Death” homogeneous recursions for a single resource queue are special cases of maximum entropy “one-step” transitions which can be applied either in an operational or stochastic context. Furthermore, an “equivalence” relationship is used to analyse two-stage cyclic queueing networks with general service times, and favourable comparisons are made with global balance and approximate results. Numerical examples provide useful information on how critically system behaviour is affected by the distributional form of interarrival and service patterns. Comments on the implication of the work to the performance analysis and aggregation of computer systems are included.

Entropy maximisation and open queueing networks with priorities and blocking

A review is carried out on the characterisation and algorithmic implementation of an extended product-form approximation, based on the principle of maximum entropy (ME), for a wide class of arbitrary finite capacity open queueing network models (QNMs) with service and space priorities. A single server finite capacity GE/GE/1/N queue with R (R Ni ≤ Ni-1, i = 2 ..., R}. The GE/GE/1/N queue is utilised, in conjunction with GE-type first two moment flow approximation formulae, as a cost-effective building block towards the establishment of a generic ME queue-by-queue decomposition algorithm for arbitrary open QNMs with space and service priorities under repetitive service blocking with random destination (RS-RD). Typical numerical results are included to illustrate the credibility of the ME algorithm against simulation for various network topologies and define experimentally pessimistic GE-type performance bounds. Remarks on the extensions of the ME algorithm to other types of blocking mechanisms, such as repetitive service blocking with fixed destination (RS-FD) and blocking-after-service (BAS), are included.

MEM for arbitrary queueing networks with multiple general servers and repetitive-service blocking

The analytic method for entropy maximisation, subject to marginal mean value constraints, is applied to characterise new product-form approximations for arbitrary FCFS (first-come-first-served) queueing networks with multiple server queues and general interarrival and service times under repetitive-service blocking involving both fixed and random destinations. For open queueing networks the maximum entropy solution suggests a decomposition into individual finite capacity multiple server queues with a censored arrival process and revised service-times. For closed queueing networks the product-form solution of open networks is initially modified to satisfy constraints on population and flow conservation and is in turn truncated and efficiently implemented via a convolution type recursive procedure with time complexity of O(M2L2), where M is the number of queues and L is the fixed population of the network. The FCFS GE/GE/c/K;N censored queue with GE (generalised exponential) interarrival-time and service-time distributions, c (c ≧ 1) multiple servers, minimum queue length, K (K ⩾ 0) and finite capacity, N (K < N < + ∞), is exactly analysed in the context of maximum entropy and used as a ‘building block’ for the approximate analysis of general queueing networks. Validation examples are presented for assessing the numerical accuracy of the maximum entropy method (MEM) and favourable comparisons against simulation results are made.

A distributed reputation and trust management scheme for mobile peer-to-peer networks

In peer-to-peer (P2P) networks, trust ratings aggregation and peer ranking are unreliable, time-consuming and space-demanding operations. The swift expansion of emerging P2P techniques towards the domain of mobile computing poses significant challenges for trust and security management. Several trust management schemes have been proposed recently to counter the security threat on P2P systems. However, due to the difficulties caused by system mobility and dynamic network topology, there is an increasing requirement of decentralized and distributed trust management schemes. In this paper, we initially investigate and analyze four typical decentralized and distributed trust management schemes. Based on the findings of this analysis, a robust distributed reputation and trust management scheme, referred to as M-trust, is proposed for mobile P2P networks. The new scheme utilizes confidence in reputation, based on interactions among peers, to reduce the computation complexity. Furthermore, distributed algorithms are presented for accurate and reliable trust ratings aggregation and space management. The performance of M-trust is evaluated in comparison to the existing schemes using extensive simulation experiments. The results demonstrate that M-trust possesses the excellent overall performance in terms of accuracy, reliability, convergence speed, and detection rate under various constraints of mobility, trust threshold and network out-degree.

Entropy maximised queueing networks with blocking and multiple job classes

Abstract The method of entropy maximisation (MEM) is applied in a state space partitioning mode for the approximation of the joint stationary queue length distribution of an M/M/1/N queue with finite capacity, N( > 1), multiple and distinct classes of jobs, R( > 1), under a complete buffer sharing scheme and mixed service disciplines drawn from the first-come-first-served (FCFS), last-come-first-served with (LCFS-PR) or without (LCFS-NPR) preemption and processor sharing (PS) rules. The marginal and aggregate maximum entropy (ME) queue length distributions and the associated blocking probabilities per class are also determined. These ME results in conjunction with the first moments of the effective flows are used, as building blocks, in order to establish a new product-form approximation for arbitrary exponential open queueing networks with multiple classes of jobs under repetitive-service (RS) blocking with random destination (RD). It is verified that the ME approximation reduces to the exact truncated solution of open multi-class reversible queueing networks. Numerical experiments demonstrate a good accuracy level of ME statistics in relation to simulation. Moreover, recent extentions of MEM for arbitrary GE-type queueing networks with RS-RD blocking and multiple classes of jobs are presented.

A maximum entropy priority approximation for a stable G/G/1 Queue

SummaryThe principle of maximum entropy is used under two different sets of mean value constraints to analyse a stableG/G/1 queue withR priority classes under preemptive-resume (PR) and non-preemptive head-of-line (HOL) scheduling disciplines. New one-step recursions for the maximum entropy state probabilities are established and closed form approximations for the marginal queue length distribution per priority class are derived. To expedite the utility of the maximum entropy solutions exact analysis, based on the generalised exponential (GE) distribution, is used to approximate the marginal mean queue length and idle state probability class constraints for both the PR and HOLG/G/1 priority queues. Moreover, these results are used as building blocks in order to provide new approximate formulae for the mean and coefficient of variation of the effective priority service-time and suggest a maximum entropy algorithm for general open queueing networks with priorities in the context of the reduced occupancy approximation (ROA) method. Numerical examples illustrate the accuracy of the proposed maximum entropy approximations in relation to simulations involving different interarrival-time and service-time distributions per class. Comments on the extension of the work to more complex types of queueing systems are included.

MEM for arbitrary closed queueing networks with RS-blocking and multiple job classes

A new product-form approximation, based on the method of entropy maximisation (MEM), is characterised for arbitrary closed queueing networks with multiple and distinct classes of jobs, Generalised Exponential (GE) service times, mixed service disciplines, complete buffer sharing and repetitive-service blocking with both fixed (RS-FD) and random destinations (RS-RD). The maximum entropy (ME) approximation implies decomposition of the network into individual multiple class GE/GE/1/N queues satisfying constraints on population and flow conservation which is, in turn, truncated and efficiently implemented by a general convolution recursive procedure for the efficient calculation of the normalising constant and typical performance metrics. A relationship between MEM and reversible closed multiple class queueing networks is identified and it is shown how the ME approximation reduces to the exact solution. Numerical validation experiments against simulation are included to demonstrate the credibility of ME results.

A Universal Maximum Entropy Algorithm for General Multiple Class Open Networks with Mixed Service Disciplines

The principle of maximum entropy (ME) is used to characterise a new product-form approximation for the analysis of arbitrary open networks of queues at equilibrium with infinite capacities, single servers, multiple job classes, distinct general exogeneous interarrival-time and service-time distributions per class, non-priority (first-come-first-served, processorsharing, last-come-first-served with or without pre or priority (preemptive-resume, non-preemptive head-of-line) service disciplines and random routing under class switching. The ME approximation suggests a decomposition of the open network into individual multiple class G/G/l queues at equilibrium with a revised arrival process for each class of jobs. A universal implementation of the ME solution is achieved by making use of the Generalised Exponential (GE) distribution to model the service and flow processes of each G/G/l queue per class. As a consequence, the ME analysis of open queueing networks can be interpreted in terms of bulkarrival and bulk-service queues with geometrically distributed bulk sizes. The credibility of the ME approximation is demonstrated by some illustrative examples and favourable comparisons with simulation and other approximate methods are made. Comments on the extension of the work to multiple server queues and general closed networks are included.

Collusion Detection and Prevention with FIRE+ Trust and Reputation Model

Recently many decentralized methods for trust management have been proposed. FIRE trust and reputation model presents a modular approach to trust and reputation from different information sources, according to availability, interaction trust, role-based trust, witness reputation, and certified reputation. However, FIRE model does not consider malicious activity and possible collusive behavior in the network nodes and is therefore susceptible to collusion attacks. In this paper we present a trust management approach to detecting collusion in direct and witness interactions among nodes based on history of interactions among nodes. Various interaction policies are defined to detect and prevent collaborative behavior in colluding nodes. Finally a multidimensional trust model FIRE+ is proposed for avoiding collusion attacks in direct and witness based interactions.