Bottleneck Station Research Articles

A Large Closed Queueing Network Containing Two Types of Node and Multiple Customer Classes: One Bottleneck Station

The paper studies a closed queueing network containing two types of node. The first type (server station) is an infinite server queueing system, and the second type (client station) is a single server queueing system with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. It is assumed that there are $r$ server stations. At the initial time moment all units are distributed in the server stations, and the $i$th server station contains $N_i$ units, $i=1,2,...,r$, where all the values $N_i$ are large numbers of the same order. The total number of client stations is equal to $k$. The expected times between departures in the client stations are small values of the order $O(N^{-1})$ ~ $(N=N_1+N_2+...+N_r)$. After service completion in the $i$th server station a unit is transmitted to the $j$th client station with probability $p_{i,j}$ ~ ($j=1,2,...,k$), and being served in the $j$th client station the unit returns to the $i$th server station. Under the assumption that only one of the client stations is a bottleneck node, i.e. the expected number of arrivals per time unit to the node is greater than the expected number of departures from that node, the paper derives the representation for non-stationary queue-length distributions in non-bottleneck client stations.

Reducing cycle times in production networks with batching stations

Batching is an important system behaviour wherein several entities in a network are combined into a single entity in all further network flow. Total cycle time can be dramatically affected by an improper choice of batch sizes. In this paper, continuous-time Markov chains are used to represent the states corresponding to three common types of batching stations: single product batching stations with one-at-a-time arrivals, single product batching stations with bulk arrival distribution, and multiple product batching stations with one-at-a-time arrivals where a common batch-processing size is used. The associated Chapman-Kolmogorov equations are solved producing surprisingly simple steady-state predictions for the batch station's steady-state behaviour in each case. Although the resulting formulas are useful alone, they are used to produce a supplement to queueing network analysis (QNA) that we call QNA[B]. The formulas are orders of magnitude more efficient computationally than conducting the corresponding simulation studies. They are validated by comparing discrete event simulation to QNA[B] in the redesign of an existing real-world electronic circuit board repair facility using design of experiments. The recommended process design for that system (the number of servers at the bottleneck station(s) and the batch size(s) at the batch-processing step(s)) cuts the current cycle time by approximately 70%.

Using storage buffer to improve unbalanced asynchronous production flow line's performance

Bottleneck station is defined as a point in the manufacturing process that holds down the amount of product which a factory can produce. It causes the inability of a system to respond to sudden changes in demand as a result of capacity restrictions. The study focuses on unbalanced lines in which one station, so called the bottleneck, has either a mean processing time or a variance larger than all other stations. By providing storage buffers, it has shown that line blocking and starving may be solved. However, the optimal placement of a predetermined amount of buffer capacity in different types of bottleneck condition is not well understood. Thus, this paper studies the relationship of bottleneck and buffer factors. These are factors that show characteristics of the bottleneck condition and buffer type (e.g. size and position). Various statistical analyses are then used to find any significance or insignificance of performance measures under each situation. Results from the experiment have revealed some interesting knowledge in allocating storage buffers for productivity improvement.

Bottleneck analysis in multiclass closed queueing networks and its application

Asymptotic behavior of queues is studied for large closed multi-class queueing networks consisting of one infinite server station with K classes and M processor sharing (PS) stations. A simple numerical procedure is derived that allows us to identify all bottleneck PS stations. The bottleneck station is defined asymptotically as the station where the number of customers grows proportionally to the total number of customers in the network, as the latter increases simultaneously with service rates at PS stations. For the case when K=M=2, the set of network parameters is identified that corresponds to each of the three possible types of behavior in heavy traffic: both PS stations are bottlenecks, only one PS station is a bottleneck, and a group of two PS stations is a bottleneck while neither PS station forms a bottleneck by itself. In the last case both PS stations are equally loaded by each customer class and their individual queue lengths, normalized by the large parameter, converge to uniformly distributed random variables. These results are directly generalized for arbitrary K=M. Generalizations for K \neq M are also indicated. The case of two bottlenecks is illustrated by its application to the problem of dimensioning bandwidth for different data sources in packet-switched communication networks. An engineering rule is provided for determining the link rates such that a service objective on a per-class throughput is satisfied.

An operation scheme for make-to-order job-shop production systems

The purpose of this paper is to solve multi-job order release problem whose objective is to minimize the sum of mean shop floor flow times (SFFTs) of orders subject to due-date constraint. A heuristic algorithm is proposed for the problem by decomposing the load profile of the shop. The load profile is divided into two parts. One is for the planning orders (new orders and unreleased orders) and the other is for the current orders being processed. The advantage of decomposing the load profile is to reduce computational complexity in the estimation step of order completion time. A job dispatching rule is also suggested to synchronize part flows in the shop floor. The experimental results show that work-in-process (WIP) in both bottleneck stations and assembly station, the SFFT of orders, and the percentage of tardy orders are significantly reduced.

The Delay of Open Markovian Queueing Networks: Uniform Functional Bounds, Heavy Traffic Pole Multiplicities, and Stability

For open Markovian queueing networks, we study the functional dependence of the mean number in the system (and thus also the mean delay since it is proportional to it by Little's Theorem) on the arrival rate or load factor. We obtain linear programs (LPs) which provide bounds on the pole multiplicity M of the mean number in the system, and automatically obtain lower and upper bounds on the coefficients {Ci} of the expansion ρCM/(1 − ρ)M + ρCM−1/(1 − ρ)M−1 + ⋯ + ρC1/(1 − ρ) + ρC0, where ρ is the load factor, which are valid for all ρ ∈ [0, 1). Our LPs can thus establish the stability of open networks for all arrival rates within capacity, while providing uniformly bounding functional expansions for the mean delay, valid for all arrival rates in the capacity region. The coefficients {Ci} can be optimized to provide the best bound at any desired value of the load factor, while still maintaining its validity for all ρ ∈ [0, 1). While the above LPs feature L(L + 1)(M + 1)/2 variables where L is the number of buffers in the network, for balanced systems we further provide a lower dimensional LP featuring just S(S + 1)/2 variables, where S is the number of stations in the network. This bound asymptotically dominates in heavy traffic a bound obtainable from the Pollaczek-Khintchine formula, and can capture interactions between multiple bottleneck stations in heavy traffic. We also provide an explicit upper bound for all scheduling policies in acyclic networks, and for the FBFS policy in open re-entrant lines.

The Throughput of Irreducible Closed Markovian Queueing Networks: Functional Bounds, Asymptotic Loss, Efficiency, and the Harrison-Wein Conjectures

Let N be the population of an irreducible closed Markovian queueing network, and denote by αu(N) the throughput of a scheduling policy u. The policy u is said to be efficient if limN→+∞αu(N) = α*, where α* is the capacity of a bottleneck station. The quantity J(u) ≔ limN→∞ N(α* − αu(N))/α* is called the asymptotic loss of u. The policy u is said to be asymptotically optimal if J(u) is as small as it can be. For multi-station irreducible closed Markovian networks, we obtain functional bounds on the throughput which are of the form αN/(N + v). The coefficients α and v are obtained by solving two linear programs (LPs), which consist of L(L + 3)/2 variables, where L is the number of buffers in the system. We are thus able to establish efficiency, as well as provide upper and lower bounds on asymptotic loss, through LP procedures. For balanced systems where all stations are equally loaded, we are able to provide reduced dimensional LPs which consist of just S(S + 1)/2 variables, where S is the number of stations in the system. We also show that the lower bound on asymptotic loss produced by the reduced dimensional LP can capture interactions between multiple bottlenecks in heavy traffic, and that it always dominates a bound obtainable by extending a conjecture of Harrison and Wein (HW) from two station systems to multi-station systems. Employing these bounds on two station systems, we prove that a certain policy developed by HW is efficient. This buffer priority policy was conjectured by them to be asymptotically optimal. They also conjectured a finite valued formula for the asymptotic loss of every buffer priority policy. This latter conjecture is not true in its full generality since there exists a certain buffer priority policy for a certain two station system which is not efficient, and consequently has infinite asymptotic loss. However, we provide credence to the conjectures by establishing that for balanced systems, the conjectured asymptotic loss of the HW-policy is a lower bound on the asymptotic loss of all policies, while the conjectured formula for the asymptotic loss applied to the exact opposite of the HW-policy is an upper bound on the asymptotic loss for all nonidling policies. The latter result is established under an additional condition identified by us which guarantees the finiteness of the asymptotic loss, and thus also the efficiency, of every nonidling scheduling policy.

Scheduling on a bottleneck station: A comprehensive cost model and heuristic algorithms

This paper presents a mathematical model for scheduling a single processor system to minimize aggregate operations cost consisting of inventory holding cost, opportunity cost due to the delay in realization of revenue, inflationary spiral effect, and the time value of money on costs yet to be incurred. We propose a number of heuristic algorithms for solving the model and provide extensive numerical experiments. Computational results are reported.

Allocation of buffers to serial production lines with bottlenecks

The optimal placement of a predetermined amount of buffer capacity in balanced serial production lines is a well-understood problem: in lines with moderate variability, the optimal allocation involves equal numbers of buffers at each site; in lines with severe variability, the equal allocation is modified slightly to place more buffers toward the center of the line. Buffering unbalanced lines is a much less well-understood problem. We study the problem of buffering serial lines with moderate variability and a single bottleneck; i.e., a single station with a larger mean processing time than all other stations. Our analysis shows that a bottleneck station draws buffers toward it, but the optimal allocation depends on the location and severity of the bottleneck, as well as the number of buffers available. Furthermore, relatively large imbalances in mean processing times are required to shift the optimal buffer allocation away from an equal allocation. Finally, line length appears to have a relatively small e...

Design and development of a stochastic high-level Petri net system for FMS performance evaluation

The main thrust of this study was to develop and implement a Petri net model for performance evaluation of FMS in the early systems design stage. A utilization estimate of each machine is essential for this stage of system design. The main purpose was to develop a methodology for obtaining utilization rate of stations in FMS. In addition, bottleneck station, throughput rate and time in system were also points of interest. In this paper, Petri nets are used as a basic tool to evaluate the performance of an FMS. How to construct Petri nets automatically is a key issue for this investigation. Since high-level Petri nets provide the power of automatic construction of an entire Petri net from the description of the modelled system, it is adopted as the basis of the FMS performance evaluation system. The advantages of high-level Petri nets are presented. Examples are provided to illustrate the mechanism of the system.

A network of priority queues in heavy traffic: One bottleneck station

In this paper we consider an open queueing network having multiple classes, priorities, and general service time distributions. In the case where there is a single bottleneck station we conjecture that normalized queue length and sojourn time processes converge, in the heavy traffic limit, to one-dimensional reflected Brownian motion, and present expressions for its drift and variance. The conjecture is motivated by known heavy traffic limit theorems for some special cases of the general model, and some conjectured “Heavy Traffic Principles” derived from them. Using the known stationary distribution of one-dimensional reflected Brownian motion, we present expressions for the heavy traffic limit of stationary queue length and sojourn time distributions and moments. For systems with Markov routing we are able to explicitly calculate the limits.

Asymptotically exact decomposition approximations for open queueing networks

Two decomposition approximations are presented for mean sojourn times in open single class queueing networks. If there is a single bottleneck station, the approximations are asymptotically exact in both light and heavy traffic. When applied to a Jackson network or an M/G/1 queue, these approximations are exact for all values of the traffic intensity. They also perform reasonably well on some small examples.

Line Balancing With Task Learning Effects

Abstract In this paper we show how the learning effects of the individual tasks can be incorporated in the initial design of the stations so that the system may not experience large idle times. We account for the learning effects in assigning tasks to stations by linking the task assignment to the formation of dynamic (in time) bottleneck stations. The station design problem is modeled as a dynamic recursive optimization with computational complexity of O( M2N), where Mis the number of tasks and Nthe number of production units. Using a computer simulation for a system designed by using our task assignment procedure, we compare the system idle times with/without ignoring the learning effects.

Effect of learning in JIT production system: A simulation experiment on microcomputer

In this study, we investigate the effect of learning in a Just-In-Time production environment. A simulation model is developed to evaluate the system performance when there is a bottleneck due to unbalance in processing times. We then analyze the situation in which the processing time at the bottleneck station can be reduced through learning. The model identifies the minimum number of Kanbans to be used at various production rates in order to attain the specified production level.

Scheduling semiconductor wafer fabrication

The impact that scheduling can have on the performance of semi-conductor wafer fabrication facilities is assessed. The performance measure considered is the mean throughput time (sometimes called cycle time, turnaround time or manufacturing interval) for a lot of wafers. A variety of input control and sequencing rules are evaluated using a simulation model of a representative, but fictitious, semiconductor wafer fabrication. Certain of these scheduling rules are derived by restricting attention to the sub-set of stations that are heavily utilized, and by using a Brownian network model, which approximates a multi-class queuing network model with dynamic control capability. Three versions of the wafer fabrication model, which differ only by the number of servers present at particular stations, are studied. The three versions have one, two, and four stations, respectively, that are heavily utilized (near 90% utilization). The simulation results indicate that scheduling has a significant impact on average throughput time, with larger improvements coming from discretionary imput control than from lot sequencing. The effects that specific sequencing rules have are highly dependent on both the type of input control used and the number of bottleneck stations in the fabrication.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>