The Distribution of Queuing Network States at Input and Output Instants

Sevcik, Kenneth C.; Mitrani, Isi

doi:10.1145/322248.322257

Cited by 238 publications

(73 citation statements)

References 11 publications

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“…Now the well-known arrival theorem for Jackson networks (see, e.g., Sevcik and Mitrani 1981) says that this probability is the same as the probability of the state seen by a job arriving at node i from node 1 in the same cyclic network but with a total population m + 1. Therefore the distribution of the time until item m + 1 is produced is the same as the distribution of the cycle time in the network with nodes i, i -1, …, and 1 and population m + 1.…”

Section: Remainingmentioning

confidence: 99%

Lead-time analysis and a control policy for production lines

Economopoulos

Kouikoglou

2011

International Journal of Production Research

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We address a control problem for a production line which produces one product to stock and faces random demand. During stockouts, the system quotes a fixed response time for each arriving order, and the customers place their orders only if the response time promised meets their deadlines. Customer orders are filled on a first come, first served basis. A penalty cost is incurred whenever a customer is served later than promised. A two-parameter admission/inventory control policy is implemented that maintains a bounded backlog and a constant inventory position (total inventory minus backlog) in the system. For production lines with exponential processing times and Poisson demand, the mean profit rate of the system is computed analytically using closed queueing network formulas. For systems with general processing or interarrival time distributions, the profit rate is estimated via simulation.Simple properties are established which ensure that the profit maximizing control parameters can be determined in finite time using exhaustive search.Numerical results show that the proposed policy performs better than the make-to-order/zero-inventory and the lost-sales/make-to-stock policies. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 IntroductionProduction control involves decisions concerning when and how much to produce and when to accept incoming orders. Such decisions have significant effects on a number of operational indices such as throughput, mean inventory and backlog levels, and quality of service. These operational indices affect the mean profit rate of the system, which comprises profit from sales and the costs associated with inventory, backlog, delays in filling orders, possible loss of sales during stockout periods and so on (see, e.g., Porteus 1990, Section 3.7, p. 615). In the literature of production control there is a rich collection of policies that minimize the inventory and backlog costs.A simple and often effective inventory control policy is one that specifies a target value for the stock level, called base stock (Zipkin 2000, Buzacott andShanthikumar 1993); the manufacturing facility produces as long as the number of finished items is smaller than the base stock and idles otherwise.In order to handle incoming orders during stockout periods, many systems apply a lost sales (LS -all incoming orders rejected) or a complete backordering (CB -all incoming orders accepted) policy. When applying LS, there is no cost of delay in filling orders but the loss of sales to competition during stockouts makes it mandatory to keep a high stock level. On the contrary, facilities under CB appear to have vast backlog queues and long delays in filling orders which, apart from high backlogging costs, could also lead to potential loss of sales due to customer impatience.An admission control policy which is analogous to the base stoc...

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Section: Remainingmentioning

confidence: 99%

Lead-time analysis and a control policy for production lines

Economopoulos

Kouikoglou

2011

International Journal of Production Research

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“…By the arrival theorem [13] , which holds for the closed product form network underlying the machine-repairman model, each first arrival sees the repairman queue in equilibrium as if its own class has never been active-so as if there are only N − 1 classes. Let W first denote the waiting time (not including service time) for a first arrival.…”

Section: The First Arrival's Waiting Timementioning

confidence: 99%

Polling with Batch Service

Boxma¹,

Wal²,

Yechiali³

2008

Stochastic Models

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“…Classical mean value analysis is based on two theorems: The Arrival Instant Distribution theorem of Lavenberg and Reiser [8] and Sevcik and Mitrani [16] and Little's Law [9]. The Arrival Instant Distribution Theorem states that "a class-r job, arriving at station i in the system with population K, sees the system with population (K-lr) in equilibrium".…”

Section: Mean Value Analysismentioning

confidence: 99%