Bucket brigades revisited: Are they always effective?

Previous work on the dynamics of bucket brigades has focused on systems in which workers can be ordered with respect to their speeds and where this ordering does not change throughout the line. While this assumption is valid in most environments, it may not be satisfied in some. We consider such environments and explore the conditions under which bucket brigades continue to be effective (compared to a traditional static allocation) with respect to self-balancing behavior and throughput performance. A two worker bucket brigade is studied where one worker is faster than the other over some part of the production line and slower over another part of the line. We analyze the dynamics and throughput of the bucket brigade in two environments with passing and blocking. We present the dynamics of the system in each region of the parameter space and provide insights and operating principles for the implementation and management of the bucket brigades under various scenarios.     

Dynamics of 2-worker bucket brigade assembly line with blocking and instantaneous walk-back

We analyze the dynamics of 2-worker m-stations bucket brigade assembly lines where the velocities of the workers on the stations are arbitrary, albeit fixed constants over each station. We provide a complete characterization of the dynamics under blocking and instantaneous walk-back.     

Two-worker blocking congestion model with walk speed m in a no-passing circular passage system

This paper introduces a blocking model and closed-form expression of two workers traveling with walk speed m (m = integer) in a no-passing circular-passage system of n stations and assuming n = m + 2, 2m + 2, …. We develop a Discrete-Timed Markov Chain (DTMC) model to capture the workers’ changes of walk, pick, and blocked states, and quantify the throughput loss from blocking congestion by deriving a steady state probability in a closed-form expression. We validate the model with a simulation study. Additional simulation comparisons show that the proposed throughput model gives a good approximation of a general-sized system of n stations (i.e., n > 2), a practical walk speed system of real number m (i.e., m ? 1), and a bucket brigade order picking application.     

Variance of the throughput of an N-station production line with no intermediate buffers and time dependent failures

In this study, the variance of the throughput of an N-station production line with no intermediate buffers and time dependent failures is analytically determined. Time to failure and time to repair distributions are assumed to be exponential. The analytical method yields a closed-form expression for the variance of the throughput. The method is based on determining the limiting variance of the total residence (sojourn) time in a specific state of an irreducible recurrent Markov process from the probability of visiting that state at time t given an initial state. This probability function is the instantaneous availability of a production system in the reliability context. A production line with no interstation buffers and time-dependent failures is basically a series system with hot standby. The same procedure can be applied to determine the variance of the throughputs of various arrangements of workstations including series, parallel, series-parallel systems provided that the instantaneous availabilities of these systems can be written explicitly. Numerical experiments show that, although the expected throughput decreases monotonically, the variance of the throughput may increase and then decrease as the number of stations in the line increases depending on the system parameters. Numerical experiments that investigate this phenomenon and also the dependence of the coefficient of variation on the number of stations are also presented in this study.     

Some limit theorems for regenerative queues

We consider a single server queue with the interarrival times and the service times forming a regenerative sequence. This traffic class includes the standard models: iid, periodic, Markov modulated (e.g., BMAP model of Lucantoni [18]) and their superpositions. This class also includes the recently proposed traffic models in high speed networks, exhibiting long range dependence. Under minimal conditions we obtain the rates of convergence to stationary distributions, finiteness of stationary moments, various functional limit theorems and the continuity of stationary distributions and moments. We use the continuity results to obtain approximations for stationary distributions and moments of an MMPP/GI/1 queue where the modulating chain has a countable state space. We extend all our results to feed-forward networks where the external arrivals to each queue can be regenerative. In the end we show that the output process of a leaky bucket is regenerative if the input process is and hence our results extend to a queue with arrivals controlled by a leaky bucket. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bucket brigades on in-tree assembly networks

In a network of subassembly lines, balance becomes more difficult to achieve as it requires that all subassembly lines be synchronized to produce at the same rate. We show how to adapt the “bucket brigade” protocol of work-sharing so that balance emerges spontaneously.     

Analysis and design of rate-based congestion control of high speed networks,I: stochastic fluid models,access regulation

The paper gives models and analytic techniques for addressing critical issues of the Broadband Integrated Services Digital Network which will use the Asynchronous Transfer Mode. The traffic is expected to be highly bursty and variable at the source and consequently a key issue is admission control. We study a 4-parameter device called a regulator which acts as a policing device as well as a traffic shaper. The device is a generalized leaky bucket with a data buffer, a token buffer supplied by a constant-rate token stream, and a peak rate controller; the outputs of the device are streams of priority and marked cells. The composite system comprising of the source and the regulator is represented in a stochastic fluid model since fluid flow has been found to have properties well matched to the ATM environment, and the Markov Modulated Fluid Source allows bursty characteristics to be accurately modelled. A complete procedure based on spectral expansions for calculating the system's stationary state distribution is given. It is shown that with proper design the regulator effectively controls a three-way trade-off between throughput, delay and burstiness. Numerical results reveal that performance is sensitive to source characteristics such as the squared coefficient of variation of burst and silent periods. The second part of the paper characterizes the output of the regulator. The distributions of the time periods spent in the various states by the output process are calculated exactly. From this an approximate Markovian characterization is obtained. The output streams of priority and marked cells are coupled to capture their correlations. For the simple case of two-state on-off sources, the approximate Markovian characterization of the regulator's output rate processes is explicitly given and it is distinguished by the property that all moments are identical to those of the actual processes. With this characterization an original goal of analyzing a composite system of access regulation and statistical multiplexing is separated, decomposed and thereby made tractable.     

An approach to credit assignment in classifier systems

We present a conceptual framework within which we can analyze simple reward schemes for classifier systems. The framework consists of a set of classifiers, a learning mechanism, and a finite automaton environment that outputs payoff. We find that many reward schemes have negative biases that degrade system performance. We analyze bucket brigade schemes, which are subgoal reward schemes, and profit sharing schemes, which aren't. By contrasting these schemes, we hope to better understand the place of subgoal reward in learning and evolution. © 1998 John Wiley & Sons, Inc.     

Stability and performance analysis of rate-based feedback flow controlled ATM networks

Motivated by the ABR class of service in ATM networks, we study a continuous-time queueing system with a feedback control of the arrival rate of some of the sources. The feedback regarding the queue length or the total workload is provided at regular intervals (variations on it, especially the EPRCA algorithm, are also considered). The propagation delays can be nonnegligible. For a general class of feedback algorithms, we obtain the stability of the system in the presence of one or more bottleneck nodes in the virtual circuit. We also obtain rates of convergence to the stationary distributions and finiteness of moments. For the single bottleneck case, we provide algorithms to compute the stationary distributions and the moments of the sojourn times in different sets of states. We also show analytically (by showing the continuity of stationary distributions and moments) that for small propagation delays, we can provide feedback algorithms which have higher mean throughput, lower probability of overflow, and lower delay jitter than any open-loop policy. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

A statistical Markov chain approximation of transient hospital inpatient inventory

Inventory levels are critical to the operations, management, and capacity decisions of inventory systems but can be difficult to model in heterogeneous, non-stationary throughput systems. The inpatient hospital is a complicated throughput system and, like most inventory systems, hospitals dynamically make managerial decisions based on short term subjective demand predictions. Specifically, short term hospital staffing, resource capacity, and finance decisions are made according to hospital inpatient inventory predictions. Inpatient inventory systems have non-stationary patient arrival and service processes. Previously developed models present poor inventory predictions due to model subjectivity, high model complexity, solely expected value predictions, and assumed stationary arrival and service processes. Also, no models present statistical testing for model significance and quality-of-fit. This paper presents a Markov chain probability model that uses maximum likelihood regression to predict the expectations and discrete distributions of transient inpatient inventories. The approach has a foundation in throughput theory, has low model complexity, and provides statistical significance and quality-of-fit tests unique to this Markov chain. The Markov chain is shown to have superior predictability over Seasonal ARIMA models.     

Inequalities for Stationary Poisson Cuboid Processes

A cuboid is a rectangular parallelepipedon. By the notion “stationary Poisson cuboid process” we understand a stationary Poisson hyperplane process which divides the Euclidean space ?^d into cuboids. It is equivalent to speak of a stationary Poisson cuboid tessellation. The distributions of volume and total edge length of the typical cuboid and the origin-cuboid of a stationary Poisson cuboid process are considered. It is shown that these distributions become minimal, in the sense of a specific order relation, in the case of quasi-isotropy. A possible connection to a more general problem, treated in [6], is also discussed.     

Stability and performance analysis of rate-based feedback flow controlled ATM networks

Motivated by ABR class of service in ATM networks, we study a continuous time queueing system with a feedback control of the arrival rate of some of the sources. The feedback about the queue length or the total workload is provided at regular intervals (variations on it, especially the traffic management specification TM 4.0, are also considered). The propagation delays can be nonnegligible. For a general class of feedback algorithms, we obtain the stability of the system in the presence of one or more bottleneck nodes in the virtual circuit. Our system is general enough that it can be useful to study feedback control in other network protocols. We also obtain rates of convergence to the stationary distributions and finiteness of moments. For the single botterneck case, we provide algorithms to compute the stationary distributions and the moments of the sojourn times in different sets of states. We also show analytically (by showing continuity of stationary distributions and moments) that for small propagation delays, we can provide feedback algorithms which have higher mean throughput, lower probability of overflow and lower delay jitter than any open loop policy. Finally these results are supplemented by some computational results. This revised version was published online in June 2006 with corrections to the Cover Date.     

Insensitivity of multiclass systems with general dynamic preemptive resume queueing disciplines

We are concerned with the insensitivity of the stationary distributions of the system states inM/G/s/m queues with multiclass customers and with LIFO preemptive resume service disciplines. We introduce general entrance and exit rules into and from waiting positions, respectively, for the behaviour of waiting customers whose service is interrupted. These rules may, roughly speaking, depend on the number of customers in the system. It is shown that the stationary distribution of the system state is insensitive not only with respect to the service time distributions but also with respect to the general entrance and exit rules. As well as the insensitivity of the service scheme, our results are obtained for a special form of state and customer type dependent arrival and service rates. Some further results are concluded related to insensitivity like the formula for the conditional mean sojourn time and the property of transformation of a Poisson input into a Poisson output by the systems.     

Stationary wave beams in strongly nonlinear three-dimensional inhomogeneous media

We derive an asymptotic expression for the evolution of stationary beams in strongly nonlinear three-dimensional media. Formulas are obtained for the distribution of the beam amplitude and phase velocity, and effectively solvable equations are constructed for the beam axial line. It is shown that with power-function nonlinearity, the determination of the beam axial line is separated from the determination of the field concentrated near this line.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 52–60, 1985.     

Probability masses fitting in the analysis of manufacturing flow lines

The analysis of manufacturing systems with finite capacity and with general service time distributions is made of two steps: the distributions have first to be transformed into tractable phase-type distributions, and then the modified system can be analytically modelled. In this paper, we propose a new alternative in order to build tractable phase-type distributions, and study its effects on the global modelling process. Called “probability masses fitting” (PMF), the approach is quite simple: the probability masses on regular intervals are computed and aggregated on a single value in the corresponding interval, leading to a discrete distribution. PMF shows some interesting properties: it is bounding, monotonic, refinable, it approximates distributions with finite support and it conserves the shape of the distribution. With the resulting discrete distributions, the evolution of the system is then exactly modelled by a Markov chain. Here, we focus on flow lines and show that the method allows us to compute upper and lower bounds on the throughput as well as good approximations of the cycle time distributions. Finally, the global modelling method is shown, by numerical experiments, to compute accurate estimations of the throughput and of various performance measures, reaching accuracy levels of a few tenths of a percent.     

An approximation for computing the throughput of closed assembly-type queueing networks

In this paper, we examine throughput (mean number of completed assemblies per unit time) of closed assembly type queueing networks where machine processing times are drawn from general distributions. The system dynamics are characterized via a set of stochastic difference equations; it is shown that the system state can be modeled by a discrete index Markov chain on a continuous state space. Standard Markovian analysis is employed to derive an approximate expression for system throughput, following discretisation of state space. Four examples of CONWIP (CONstant Work IN Process) systems are given that illustrate the results.     

Straight chains of interconnected mass points under the action of non-symmetric dry friction

The motion of a chain of interconnected equal material points placed on a rough straight line and connected by equal kinematical constraints is considered. It is supposed that the system experiences a small non-symmetric Coulomb dry frictional force acting from the line side upon each mass point and opposite to the direction of motion. The magnitude of this force depends on the direction of motion. To the equation of motion the procedure of averaging is applied. The expression for the stationary “on the average” velocity of the motion of the system as a single whole is found. The obtained theoretical results can be used for the design of worm-like vibration robots. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Queueing Networks of Random Link Topology: Stationary Dynamics of Maximal Throughput Schedules

In this paper, we study the stationary dynamics of a processing system comprised of several parallel queues and a single server of constant rate. The connectivity of the server to each queue is randomly modulated, taking values 1 (connected) or 0 (severed). At any given time, only the currently connected queues may receive service. A key issue is how to schedule the server on the connected queues in order to maximize the system throughput. We investigate two dynamic schedules, which are shown to stabilize the system under the highest possible traffic load, by scheduling the server on the connected queue of maximum backlog (workload or job number). They are analyzed under stationary ergodic traffic flows and connectivity modulation. The results also extend to the more general case of random server rate.We then investigate the dynamics of acyclic (feed-forward) queueing networks with nodes of the previous type. Their links (connectivities) are stochastically modulated, inducing fluctuating network topologies. We focus on the issue of network throughput and show that it is maximized by simple node server schedules. Rate ergodicity of the traffic flows traversing the network is established, allowing the computation of the maximal throughput.Queueing networks of random topology model several practical systems with unreliable service, including wireless communication networks with extraneous interference, flexible manufacturing systems with failing components, production management under random availability of resources etc.Research supported in part by the National Science Foundation.This revised version was published online in June 2005 with corrected coverdate     

Sensitivity analysis and decomposition of unreliable production lines with blocking

The analysis of finitebuffered, unreliable production lines is often based on the method of decomposition, where the original system is decomposed into a series of twostage subsystems that can be modeled as quasi birthdeath processes. In this paper, we present methods for computing the gradients of the equilibrium distribution vector for such processes. Then we consider a specific production line with finite buffers and machine breakdowns and develop an algorithm that incorporates gradient estimation into the framework of Gershwin's approximate decomposition. The algorithm is applied to the problem of workforce allocation to the machines of a production line to maximize throughput. It is shown that this problem is equivalent to a convex mathematical programming problem and, therefore, a globally optimal solution can be obtained.     

Unsteady MHD Couette flow in a rotating system

The unsteady hydromagnetic Couette flow of a viscous incompressible electrically conducting fluid in a rotating system has been considered. An exact solution of the governing equations has been obtained by using a Laplace transform. Solutions for the velocity distributions as well as shear stresses have been obtained for small times as well as for large times. It is found that for large times the primary velocity decreases with increase in the rotation parameter K² while it increases with increase in the magnetic parameter M². It is also found that with increase in K², the secondary velocity v₁ decreases near the stationary plate while it increases near the moving plate. On the other hand, the secondary velocity decreases with increase in the magnetic parameter.