Burke's Theorem Research Articles

Performance measures of parallel tandem open queueing network

PurposeThe authors consider parallel four-state tandem open queueing network. The queue capacity is infinite. Passenger arrival rate is Poisson distribution and service rate is exponential distribution. The queue is constructed in the form of tandem queue, and each and every queue of tandem queue is single server (M/M/1) queue. In tandem queue, passengers will leave the system once they receive service from both the states. The purpose of this paper is to provide performance analysis for four-state tandem open queue network, and a governing equation is formulated with the help of transition diagram. Using Burke theorem, the authors formulated equation for average number of passenger in the system, average waiting time of passenger in the system, average number of passenger in the queue and average waiting time of passenger in the queue.Design/methodology/approachThis paper used Burke’s theorem.FindingsIn this paper, performance analysis is done for parallel four-state tandem open queueing network and performance measure solved using Burkes theorem formula. K. Sreekanth et al. has done performance analysis for single tandem queue with three states. In this paper, the authors have done performance analysis for two tandem queues parallel with four states. This four-state tandem open queueing network is suitable for real world applications. This paper can extend for more number of service states and multi-server states according to the application, and in such case, the authors have to prove and explain with numerical examples. This analysis is more useful for the applications such as airports, railway stations, bus-stands and banks.Originality/valueIn this paper, parallel four-state tandem open queueing network and performance measure has been solved using Burke’s theorem formula.

Multipath Delay Analysis Using Queuing Theory for Gateway Selection in Hybrid MANET

Hybrid Mobile ad hoc network (H-MANET) is a network in which MANET is connected to the Internet using some special nodes called Internet Gateways (IGs). Previous work in this field considered the infinite buffer scenario and therefore buffer overflow would never occur. But in realistic MANET, the buffer size of each node is strictly bounded that incurs packet loss and intensifies delay. In this paper, a mathematical model has been proposed for modeling of multipath delay analysis for IG selection in H-MANET. The proposed work is used to estimate end to end delay of each individual path amongst Source Node (SN) and IGs. The M/M/m/R network model has been employed for modeling the network in which packets arrive in Poisson distribution in a bounded queue with finite buffer capacity. Burke theorem has been used to calculate queuing delay of each individual path as well as the complete path from the SN to an IG. Furthermore, best IG is selected for data transmission based on delay analysis of multiple paths. The Opnet modeler 18.0 has been used to perform the simulation work. The numerical results represent the effectiveness of proposed model.

On the universality of the nonstationary ideal

AbstractBurke proved that the generalized nonstationary ideal, denoted by NS, is universal in the following sense: every normal ideal, and every tower of normal ideals of inaccessible height, is a canonical Rudin‐Keisler projection of the restriction of NS to some stationary set. We investigate how far Burke's theorem can be pushed, by analyzing the universality properties of NS with respect to the wider class of ‐systems of filters introduced by Audrito and Steila. First we answer a question of Audrito and Steila, by proving that ‐systems of filters do not capture all kinds of set‐generic embeddings. We provide a characterization of supercompactness in terms of short extenders and canonical projections of NS, without any reference to the strength of the extenders; as a corollary, NS can consistently fail to canonically project to arbitrarily strong short extenders. We prove that ω‐cofinal towers of normal ultrafilters, e.g., the kind used to characterize I2 and I3 embeddings, are well‐founded if and only if they are canonical projections of NS. Finally, we provide a characterization of “ is Jónsson” in terms of canonical projections of NS.

Mobile Access Coordinated Wireless Sensor Networks—Design and Analysis

This paper introduces and analyzes a novel mobile access (MA) coordinated wireless sensor network (MC-WSN) architecture. In conventional sensor networks with mobile access points (SENMA), the MA points traverse the network to collect information directly from individual sensors. While simplifying the routing process, a major limitation with SENMA is that data transmission is limited by the physical speed of the MAs and their trajectory length, resulting in low throughput and large delay. In an effort to resolve this problem, we introduce the MC-WSN architecture, for which a major feature is that: through active network deployment and topology design, the number of hops from any sensor to the MA can be limited to a prespecified number. In this paper, first, we discuss the optimal topology design for MC-WSN such that the average number of hops between the source and its nearest sink is minimized. Second, we calculate the throughput of MC-WSN and illustrate the effect of the number of hops on the throughput. Third, we establish the queuing model for the cluster heads based on the Kleinrock independence assumption and Burke's theorem, and show that the traffic at each cluster head can be modeled as an independent M/M/1 queue. Finally, based on the queue modeling, we provide an in-depth analysis on network stability, delay, and energy efficiency. Our analysis is demonstrated through numerical results. It is shown that under stable system conditions, MC-WSN achieves a much higher throughput and considerably lower delay and energy consumption over SENMA.

Modeling of an Efficient Integration of MANET and Internet Using Queuing Theory

Integrating mobile adhoc network (MANET) to the Internet is a burning issue due to its various advantages over standalone MANET. Such integration provides many usage scenarios for both fixed network and MANET together like conference meeting, wireless classroom and information processing on the move such as in trains and airport. Some of the nodes in MANET called Internet gateways (IGWs) act as an interface between MANET and fixed network such as Internet which offer the Internet connectivity to the mobile host. Such integrated network is called hybrid MANET. A mathematical model has been proposed in this paper to calculate end to end (ETE) delay and throughput between active source and each IGW node by splitting the traffic in multiple paths. The proposed model follows M/M/m queuing network and Burke theorem has been used to calculate ETE delay of multiple paths between active source (AS) and each IGW. Further, through analysis the best Internet gateway is selected based on minimum ETE delay between AS and each IGW for data delivery to the Internet. The proposed model is useful for analyzing and selecting IGW out of multiple IGWs in hybrid MANET. Simulation results show the validity of the proposed approach.

Convergence of tandem Brownian queues

AbstractIt is known that in a stationary Brownian queue with both arrival and service processes equal in law to Brownian motion, the departure process is a Brownian motion, identical in law to the arrival process: this is the analogue of Burke's theorem in this context. In this paper we prove convergence in law to this Brownian motion in a tandem network of Brownian queues: if we have an arbitrary continuous process, satisfying some mild conditions, as an initial arrival process and pass it through an infinite tandem network of queues, the resulting process weakly converges to a Brownian motion. We assume independent and exponential initial workloads for all queues.

A review of Burke’s theorem for Brownian motion

Burke's theorem is a well-known fundamental result in queueing theory, stating that a stationary M/M/1 queue has a departure process that is identical in law to the arrival process and, moreover, for each time t, the following three random objects are independent: the queue length at time t, the arrival process after t and the departure process before t. Burke's theorem also holds for a stationary Brownian queue. In particular, it implies that a certain "complicated" functional derived from two independent Brownian motions is also a Brownian motion. The aim of this overview paper is to present an independent complete explanation of this phenomenon.

Multipath routing in mobile ad hoc network with probabilistic splitting of traffic

Multipath routing is a burning issue in mobile ad hoc network due to its various advantages over single path routing. Some of these advantages are load balancing, bandwidth aggregation, and fault tolerance. Multipath routing means multiple paths exist between source and destination pair. Many works discussed in section 2 addressed queuing delays, but none of them suggested queuing delay for multiple path deliveries of data in mobile ad hoc network context. In this paper, we have designed a mathematical model to compute delay and throughput for multipath. Our model follow the network of M/M/1 queues, and we have applied Burke's theorem to calculate the queuing delay of the packet in mobile network scenario. This model can be used to estimate delay and throughput of an individual path. Further, through the analysis the best path for data delivery out of available multiple paths as well as the multipath path can be used simultaneously for data delivery to the destination. Simulation result shows that splitted traffic multiple paths outperform splitted traffic. Therefore, our model is useful for design and analysis of ad hoc network. The simulation work has been carried out in Qualnet simulator.

A Relation between Queue-Length Distributions during Server Vacations in Queues with Batch Arrivals, Batch Services, or Multiclass Arrivals: An Extension of Burke's Theorem

This paper presents a relation between queue-length distributions during vacations and those at vacation start and end times, in M/G/1 vacation queues with batch arrivals, batch services or multi-class arrivals. In order to obtain the relation, we develop a new approach that extends Burke's theorem to queues with batch arrivals, batch services or multi-class services. We also give an example to demonstrate how to apply the proposed relation to the analysis of such a queue.

Scaling for a one-dimensional directed polymer with boundary conditions

We study a (1+1)-dimensional directed polymer in a random environment on the integer lattice with log-gamma distributed weights. Among directed polymers, this model is special in the same way as the last-passage percolation model with exponential or geometric weights is special among growth models, namely, both permit explicit calculations. With appropriate boundary conditions, the polymer with log-gamma weights satisfies an analogue of Burke's theorem for queues. Building on this, we prove the conjectured values for the fluctuation exponents of the free energy and the polymer path, in the case where the boundary conditions are present and both endpoints of the polymer path are fixed. For the polymer without boundary conditions and with either fixed or free endpoint, we get the expected upper bounds on the exponents.

The M/M/1 queue is Bernoulli

The classical output theorem for the M/M/1 queue, due to Burke (1956), states that the departure process from a stationary M/M/1 queue, in equilibrium, has the same law as the arrivals process, that is, it is a Poisson process. We show that the associated measure-preserving transformation is metrically isomorphic to a two-sided Bernoulli shift. We also discuss some extensions of Burke's theorem where it remains an open problem to determine if, or under what conditions, the analogue of this result holds.

Queues, stores, and tableaux

Consider the single-server queue with an infinite buffer and a first-in–first-out discipline, either of type M/M/1 or Geom/Geom/1. Denote by 𝒜 the arrival process and by s the services. Assume the stability condition to be satisfied. Denote by 𝒟 the departure process in equilibrium and by r the time spent by the customers at the very back of the queue. We prove that (𝒟, r) has the same law as (𝒜, s), which is an extension of the classical Burke theorem. In fact, r can be viewed as the sequence of departures from a dual storage model. This duality between the two models also appears when studying the transient behaviour of a tandem by means of the Robinson–Schensted–Knuth algorithm: the first and last rows of the resulting semistandard Young tableau are respectively the last instant of departure from the queue and the total number of departures from the store.

Queues, stores, and tableaux

Consider the single-server queue with an infinite buffer and a first-in–first-out discipline, either of type M/M/1 or Geom/Geom/1. Denote by 𝒜 the arrival process and by s the services. Assume the stability condition to be satisfied. Denote by 𝒟 the departure process in equilibrium and by r the time spent by the customers at the very back of the queue. We prove that (𝒟, r) has the same law as (𝒜, s), which is an extension of the classical Burke theorem. In fact, r can be viewed as the sequence of departures from a dual storage model. This duality between the two models also appears when studying the transient behaviour of a tandem by means of the Robinson–Schensted–Knuth algorithm: the first and last rows of the resulting semistandard Young tableau are respectively the last instant of departure from the queue and the total number of departures from the store.

Entropy and the timing capacity of discrete queues

Queueing systems which map Poisson input processes to Poisson output processes have been well-studied in classical queueing theory. This paper considers two discrete-time queues whose analogs in continuous-time possess the Poisson-in-Poisson-out property. It is shown that when packets arriving according to an arbitrary ergodic stationary arrival process are passed through these queueing systems, the corresponding departure process has an entropy rate no less (some times strictly more) than the entropy rate of the arrival process. Some useful by-products are discrete-time versions of: (i) a proof of the celebrated Burke's (1956) theorem, (ii) a proof of the uniqueness, amongst renewal inputs, of the Poisson process as a fixed point for exponential server queues proposed by Anantharam (1993), and (iii) connections with the timing capacity of queues described by Anantharam and Verdu (1996).

Joint Burke's Theorem and RSK Representation for a Queue and a Store

Consider the single server queue with an infinite buffer and a FIFO discipline, either of type M/M/1 or Geom/Geom/1. Denote by $\mathcal{A}$ the arrival process and by $s$ the services. Assume the stability condition to be satisfied. Denote by $\mathcal{D}$ the departure process in equilibrium and by $r$ the time spent by the customers at the very back of the queue. We prove that $(\mathcal{D},r)$ has the same law as $(\mathcal{A},s)$ which is an extension of the classical Burke Theorem. In fact, $r$ can be viewed as the departures from a dual storage model. This duality between the two models also appears when studying the transient behavior of a tandem by means of the RSK algorithm: the first and last row of the resulting semi-standard Young tableau are respectively the last instant of departure in the queue and the total number of departures in the store.

Brownian analogues of Burke's theorem

We discuss Brownian analogues of a celebrated theorem, due to Burke, which states that the output of a (stable, stationary) M/M/1 queue is Poisson, and the related notion of quasireversibility. A direct analogue of Burke's theorem for the Brownian queue was stated and proved by Harrison (Brownian Motion and Stochastic Flow Systems, Wiley, New York, 1985). We present several different proofs of this and related results. We also present an analogous result for geometric functionals of Brownian motion. By considering series of queues in tandem, these theorems can be applied to a certain class of directed percolation and directed polymer models. It was recently discovered that there is a connection between this directed percolation model and the GUE random matrix ensemble. We extend and give a direct proof of this connection in the two-dimensional case. In all of the above, reversibility plays a key role.

Exponential Waiting Time for a Big Gap in a One-Dimensional Zero-Range Process

The first time that the $N$ sites to the right of the origin become empty in a one-dimensional zero-range process is shown to converge exponentially fast, as $N \rightarrow \infty$, to the exponential distribution, when divided by its mean. The initial distribution of the process is assumed to be one of the extremal invariant measures $\nu_\rho, \rho \in (0, 1)$, with density $\rho/(1 - \rho)$. The proof is based on the classical Burke theorem.

Probability and Random Processes for Electrical Engineering

Probability and Random Processes for Electrical Engineering

Burke's theorem on passage times in Gordon–Newell networks

In a closed cycle of exponential queues where the first and the last nodes are multiserver queues while the other nodes are single-server queues, the cycle-time distribution has a simple product form. The same result holds for passage-time distributions on overtake-free paths in Gordon–Newell networks. In brief, we prove Burke's theorem on passage times in closed networks.

Burke's theorem on passage times in Gordon–Newell networks

In a closed cycle of exponential queues where the first and the last nodes are multiserver queues while the other nodes are single-server queues, the cycle-time distribution has a simple product form. The same result holds for passage-time distributions on overtake-free paths in Gordon–Newell networks. In brief, we prove Burke's theorem on passage times in closed networks.