Secret sharing schemes with partial broadcast channels

In a conventional secret sharing scheme a dealer uses secure point-to-point channels to distribute the shares of a secret to a number of participants. At a later stage an authorised group of participants send their shares through secure point-to-point channels to a combiner who will reconstruct the secret. In this paper, we assume no point-to-point channel exists and communication is only through partial broadcast channels. A partial broadcast channel is a point-to-multipoint channel that enables a sender to send the same message simultaneously and privately to a fixed subset of receivers. We study secret sharing schemes with partial broadcast channels, called partial broadcast secret sharing schemes. We show that a necessary and sufficient condition for the partial broadcast channel allocation of a (t, n)-threshold partial secret sharing scheme is equivalent to a combinatorial object called a cover-free family. We use this property to construct a (t, n)-threshold partial broadcast secret sharing scheme with O(log n) partial broadcast channels. This is a significant reduction compared to n point-to-point channels required in a conventional secret sharing scheme. Next, we consider communication rate of a partial broadcast secret sharing scheme defined as the ratio of the secret size to the total size of messages sent by the dealer. We show that the communication rate of a partial broadcast secret sharing scheme can approach 1/O(log n) which is a significant increase over the corresponding value, 1/n, in the conventional secret sharing schemes. We derive a lower bound on the communication rate and show that for a (t,n)-threshold partial broadcast secret sharing scheme the rate is at least 1/t and then we propose constructions with high communication rates. We also present the case of partial broadcast secret sharing schemes for general access structures, discuss possible extensions of this work and propose a number of open problems.      

A mathematical model for discrete-time buffer systems with correlated output process

This article considers an infinite buffer system with one or more input channels and multiple output channels. Transmission of messages from the buffer is synchronous and the arrival process of messages to the buffer is general. Each of the output channels is subjected to a random interruption process, i.e., the number of available output channels varies in time and is stochastic. The analysis of this system is carried out under the assumption that the output process can be described as a first order Markov process, i.e., the probability distribution of the number of available output channels during some clock time interval depends only on the number of available output channels during the previous clock time interval.A set of equations describing the behavior of this buffer system is derived. For a number of interesting special cases this set is solved and explicit expressions are obtained for the probability generating function of the number of messages in the buffer. Several prior studies are found as special cases of the present one. An illustrative example is treated and the results are compared to the ones obtained for an uncorrelated output process with the same equilibrium distribution. Some considerable deviations from these results are found.     

M/G/1/MLPS compared to M/G/1/PS

Multilevel processor sharing scheduling disciplines have recently been resurrected in papers that focus on the differentiation between short and long TCP flows in the Internet. We prove that, for M/G/1 queues, such disciplines are better than the processor sharing discipline with respect to the mean delay whenever the hazard rate of the service time distribution is decreasing.     

A note on the stability of the many-queue head-of-the-line processor-sharing system with permanent customers

We consider a single server system consisting of n queues with different types of customers and k permanent customers. The permanent customers and those at the head of the queues are served in processor-sharing by the service facility (head-of-the-line processor-sharing). By means of Loynes’ monotonicity method a stationary work load process is constructed and using sample path analysis general stability conditions are derived. They allow to decide which queues are stable and, moreover, to compute the fraction of processor capacity devoted to the permanent customers. In case of a stable system the constructed stationary state process is the only one and for any initial state the system converges pathwise to the steady state. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Stochastic Bounds for Monotonic Processor Sharing Networks

We consider a network of processor sharing nodes with independent Poisson arrival processes. Nodes are coupled through their service capacity in that the speed of each node depends on the number of customers present at this and any other node. We assume the network is monotonic in the sense that removing a customer from any node increases the service rate of all customers. Under this assumption, we give stochastic bounds on the number of customers present at any node. We also identify limiting regimes that allow to test the tightness of these bounds. The bounds and the limiting regimes are insensitive to the service time distribution. We apply these results to a number of practically interesting systems, including the discriminatory processor sharing queue, the generalized processor sharing queue, and data networks whose resources are shared according to max–min fairness.     

Fluid and diffusion limits for transient sojourn times of processor sharing queues with time varying rates

We provide an approximate analysis of the transient sojourn time for a processor sharing queue with time varying arrival and service rates, where the load can vary over time, including periods of overload. Using the same asymptotic technique as uniform acceleration as demonstrated in [12] and [13], we obtain fluid and diffusion limits for the sojourn time of the M_t/M_t/1 processor-sharing queue. Our analysis is enabled by the introduction of a “virtual customer” which differs from the notion of a “tagged customer” in that the former has no effect on the processing time of the other customers in the system. Our analysis generalizes to non-exponential service and interarrival times, when the fluid and diffusion limits for the queueing process are known.     

The effects of service disciplines on the performance of a nonreliable terminal system

In this paper, we consider a stochastic queueing model for the performance evaluaton of a real-life computer system consisting of n terminals connected with a CPU. A user at terminal i has thinking and processing time depending on the index i. Let us suppose that the operational system is subject to random breakdowns, which may be software or hardware ones, stopping the service both at the terminals and at the CPU. The failure-free operation times of the system and the restoration times are random variables. Busy terminals are also subject to random breakdowns not affecting the system operation. The failure-free operation times and the repair times of a busy terminal i are random variables with distribution function depending on index i. The breakdowns are serviced by a single repairman providing preemptive priority to the system's failure, while the restoration at the terminals are carried out according to the FIFO rule. We assume that each user generates only one job at a time, and he waits at the CPU before he starts thinking again, that is, the terminal is inactive while waiting at the CPU, and it cannot break down. All random variables involved in the model construction are assumed to be exponentially distributed and independent of each other. The aim of this paper is to investigate the effect of different service disciplines, such as FIFO, processor sharing, priority processor sharing, and polling, on the main performance measures, such as utilizations, response times, throughput, and mean queue length. Supported by the Hungarian National Foundation for Scientific Research (grant No. ORKA T014974/95). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló. Hungary, 1997, Part II.     

Processor-sharing queues: Some progress in analysis

This paper reviews some recent results based on new techniques used in the analysis of main processor-sharing queueing systems. These results include the solutions of the problems of determining the sojourn time distributions and the distributions of the number of jobs in the M/G/1/t8 queue under egalitarian and feedback (foreground-background) processor-sharing disciplines. A brief discussion of some related results is also given.     

M/G/1/MLPS compared with M/G/1/PS within service time distribution class IMRL

Multilevel processor-sharing (MLPS) disciplines were originally introduced by Kleinrock (in computer applications 1976) but they were forgotten for years. However, due to an application related to the service differentiation between short and long TCP flows in the Internet, they have recently gained new interest. In this paper we show that, if the service time distribution belongs to class IMRL, the mean delay in the M/G/1 queue is reduced when replacing the PS discipline with any MLPS discipline for which the internal disciplines belong to {FB, PS}. This is a generalization of our earlier result where we restricted ourselves to the service time distribution class DHR, which is a subset of class IMRL.     

The optimal allocation of server time slots over different classes of patients

We present a model for assigning server time slots to different classes of patients. The objective is to minimize the total expected weighted waiting time of a patient (where different patient classes may be assigned different weights). A bulk service queueing model is used to obtain the expected waiting time of a patient of a particular class, given a feasible allocation of service time slots. Using the output of the bulk service queueing models as the input of an optimization procedure, the optimal allocation scheme may be identified. For problems with a large number of patient classes and/or a large number of feasible allocation schemes, a step-wise heuristic is developed. A common example of such a system is the allocation of operating room time slots over different medical disciplines in a hospital.     

A simulation tool to evaluate the performance of finite-source queueing models

This paper deals with a software tool to evaluate the main characteristics of a nonhomogeneous finite-source queueing model to describe the performance of a multi-terminal system subject to random breakdowns under FIFO, priority processor sharing, and polling service disciplines. The model studied here is actually a closed queueing network network with three nonindependent service stations (CPU, terminals, and repairman), and a finite number of customers (jobs), which have different service rates at the service stations. The aim of this paper is to introduce the FQM (finite-source queueing model) program package, which was developed at the Institute of Mathematics and Informatics of Lajos Kossuth University in Debrecen, Hungary, and to investigate the performance of the above-mentioned finite-source queueing models. At the end we give a sample result to illustrate the problem in question. Supported by the Hungarian National Foundation for Scientific Research (grant Nos. OTKA T014974/95 and T016933/95). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

On the distribution of the number of customers in the symmetric M/G/1 queue

We consider an M/G/1 queue with symmetric service discipline. The class of symmetric service disciplines contains, in particular, the preemptive last-come-first-served discipline and the processor-sharing discipline. It has been conjectured in Kella et al. [1] that the marginal distribution of the queue length at any time is identical for all symmetric disciplines if the queue starts empty. In this paper we show that this conjecture is true if service requirements have an Erlang distribution. We also show by a counterexample, involving the hyperexponential distribution, that the conjecture is generally not true. AMS Subject Classifications Primary—60K25; Secondary—90B22     

On the transient behavior of the processor sharing queue

We analyze the transient behavior of the single server queue under the processor sharing discipline. Under fairly general assumptions, we give the rate of growth of the number of customers in the queue as well as the asymptotic behavior of the residual service times described in terms of a renormalized point process.     

Dynamic power control in a fading downlink channel subject to an energy constraint

A central controller chooses a state-dependent transmission rate for each user in a fading, downlink channel by varying transmission power over time. For each user, the state of the channel evolves over time according to an exogenous continuous-time Markov chain (CTMC), which affects the quality of transmission. The traffic for each user, arriving at the central controller, is modeled as a finite-buffer Markovian queue with adjustable service rates. That is, for each user data packets arrive to the central controller according to a Poisson process and packet size is exponentially distributed; an arriving packet is dropped if the associated buffer is full, which results in degradation of quality of service. The controller forwards (downlink) the arriving packets to the corresponding user according to an optimally chosen transmission rate from a fixed set A _i of available values for each user i, depending on the backlog in the system and the channel state of all users. The objective is to maximize quality of service subject to an upper bound on the long-run average power consumption. We show that the optimal transmission rate for each user is solely a function of his own packet queue length and channel state; the dependence among users is captured through a penalty rate. Further, we explicitly characterize the optimal transmission rate for each user. This project is partially supported by Motorola grant # 0970-350-AF24. The authors thank Phil Fleming,Randy Berry and Achal Bassamboo for helpful comments.     

TheM/G/1 queue with processor sharing and its relation to a feedback queue

The central model of this paper is anM/M/1 queue with a general probabilistic feedback mechanism. When a customer completes his ith service, he departs from the system with probability 1–p(i) and he cycles back with probabilityp(i). The mean service time of each customer is the same for each cycle. We determine the joint distribution of the successive sojourn times of a tagged customer at his loops through the system. Subsequently we let the mean service time at each loop shrink to zero and the feedback probabilities approach one in such a way that the mean total required service time remains constant. The behaviour of the feedback queue then approaches that of anM/G/1 processor sharing queue, different choices of the feedback probabilities leading to different service time distributions in the processor sharing model. This is exploited to analyse the sojourn time distribution in theM/G/1 queue with processor sharing.Some variants are also considered, viz., anM/M/1 feedback queue with additional customers who are always present, and anM/G/1 processor sharing queue with feedback.     

On stability of collaborative supplier selection

This note discusses the possibility of fair gain sharing in cooperative situations where players optimally partition themselves across a number of alternative channels. An example is group purchasing among a set of buyers facing with a range of suppliers. We introduce channel selection games as a new class of cooperative games and give a representation of their cores. With two channels (suppliers), the game has a non-empty core if the gain functions across every individual channel is supermodular.     

Reliability analysis of complex communication systems

This paper deals with a first-come, first-served (FCFS) queueing model to analyze the asymptotic behavior of a heterogeneous finite-source communication system with a single processor. Each source and the processor are assumed to operate in independent random environments, allowing the arrival and service processes to be Markov-modulated ones. Each message is characterized by its own exponentially distributed source and processing time with parameter, depending on the state of the corresponding environment, that is, the arrival and service rates are subject to random fluctuations. Assuming that the arrival rates of the messages are many times greater than their service rates (“fast” arrival), it is shown that the time to the first system failure converges in distribution, under appropriate norming, to an exponentially distributed random variable. Some simple examples are considered to illustrate the effectiveness of the method proposed by comparing the approximate results to the exact ones. Supported by the Hungarian National Foundation for Scientific Research (grant No. OTKA T14974/95). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Frequency assignment in a SDMA satellite communication system with beam decentring feature

In satellite communication, Spatial Division Multiple Access (SDMA) has become one of the most promising techniques that can accommodate continuing increase in the number of users and traffic demands. The technology is based on radio resource sharing that separates communication channels in space. It relies on adaptive and dynamic beam-forming technology and well-designed algorithms for resource allocation among which frequency assignment is considered. This paper studies static Frequency Assignment Problem (FAP) in a satellite communication system involving a satellite and a number of users located in a service area. The objective is to maximize the number of users that the system can serve while maintaining the signal to interference plus noise ratio of each user under a predefined threshold. Traditionally, interference is treated as fixed (binary interferences or fixed minimal required separation between frequencies) . In this paper, the interference is cumulative and variable. To solve the problem, we work on both discrete and continuous optimizations. Integer linear programming formulations and greedy algorithms are proposed for solving the discrete frequency assignment problem. The solution is further improved by beam decentring algorithm which involves continuous adjustment of satellite beams and deals with non-linear change of interference.     

Mean delay analysis for a message priority-based polling scheme

We consider a multi-access communication channel such as a centrally-controlled polling system, a distributed token-based ring, or a bus network. A message priority-based polling procedure is used to control the access to the channel. This procedure requires the server to have no advance information concerning the number of messages resident at a station prior to its visit to the station. Messages arriving at each station belong to one of two priority classes: class-1 (high priority) and class-2 (low priority). Class-1 messages are served under an exhaustive service discipline, while class-2 messages are served under a limited service discipline. Class-1 messages have non-preemptive priority over class-2 messages resident at the same station. Using a fully symmetric system model, an exact expression for the sum of the mean waiting times of class-1 and class-2 messages is first derived. Upper and lower bounds for the mean message waiting times for each individual message class are then obtained.This work was supported by NFS Grant No. NCR-8914690, Pacific-Bell and MICRO Grant No. 90-135 and US West Contract No. D890701.