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**Introduction**

**Introduction**

In many industrial processes, the service is interrupted because of the occurrence of breakdown in the facility that provides the service. The entities will not be serviced unless the facility is repaired. The server if human, may be in need of rest from time to time (Yadavalli et al [44]) or if non-human may be subject to two modes of failure, partial or total. That is, when the service facility is in partial failure mode, it gives service with a lower rate than in normal operating conditions. Various authors have analysed queueing systems where the service facility is subject to two modes of failure (Madan [49], Jain and Sharma [50], Reddy [51], and Sridharan and Jayashree [52]). Queueing systems with two modes of failure and arrivals and services in batches have not been considered so far. Such types of service interruptions are common in industry, factories, telephone booths and in operation of mechanical devices such as electronic computers, etc. In this chapter an M/M/1 queueing system is considered where the service facility is subject to two modes of failure, arrivals are in batches of varying size and service is rendered for batches of fixed size.

**Model description**

In this model units arrive at the system in batches of varying size and batches are pre-ordered for service purposes. The service of units is rendered in batches of fixed size and the service times of successive batches are distributed exponentially by a single server with rate µ1 in normal working condition and at a slower rate µ2 ,( µ2 < µ1 ) in case of partial failure of the service channel. One of the underlying assumptions about the repair process is that it starts instantaneously. If the service channel repair in the partial failure mode is complete, the unit enters the normal working mode; otherwise it goes to the failure mode.

**The general modelling approach**

**The general modelling approach**

Modelling a completely random arrival process traditionally involves using the Poisson distribution (negative exponentially distributed inter-arrival times) as the cornerstone of analysis in generating an ordered sequence of arrival events. This implies that the arrival system is treated as being Markovian. If arrivals are considered to occur within a temporal sequence of equal time intervals, the cumulative Poisson distribution can adequately generate arrivals with the passage of time. The Poisson distribution of arrivals is given by n! e P n n λ λ − = n=0,1,2… and λ > 0 (6.1) where n = no. of arrivals in a given time interval λ = average no. of arrivals in the temporal sequence of time intervals An example of the generation of a Poisson based arrival process for λ = 8 over 200 one minute time intervals is shown in Fig. 6.1.1. The generation of the arrival process is driven by a random number generator. The adequacy of the generation process is demonstrated by the achieved results.

**The general modelling approach**

In a similar fashion to the modelling of completely random arrivals (See par. 6.1.1), the modelling of a single completely random service process often involves the Poisson distribution (negative exponentially distributed service times) in generating an ordered sequence of service events. This implies that the service system is treated as being Markovian. If consecutive service events are considered to occur within a temporal sequence of equal time intervals (synchronously identical to the arrival time intervals) the cumulative Poisson distribution can adequately generate service events with the passage of time. The Poisson distribution of service events is given by n! e P n n µ µ − = n=0,1,2… and µ > 0 (6.2) where n = no. of service events offered in a given time interval µ = average no. of service events offered in the temporal sequence of time intervals An example of the generation of the service process for µ = 10 over 200 one minute time intervals is shown in Fig. 6.1.2 The generation of the service process is driven by a random number generator. The adequacy of the generation process is demonstrated by the achieved results.

**A model of the events which take place within a time interval is an example of**

**A model of the events which take place within a time interval is an example of**

a highly simplified model of a deterministic instantaneous replenishment inventory system which allows shortages to occur during the time interval i.e. when some service events are analogously on offer but not used within the interval as a result of insufficient arrivals. One may speculate that such an elementary model does not meet the requirement of mathematical elegance, or that an attempt is being made to approach the modelling problem pragmatically to avoid immersion into higher mathematics. At this juncture of the modelling process one should await the results which follow, results which are based on further development of the system modelling approach before prematurely judging the merit of the model. The resulting orbit of number of entities in the system which is obtained by merging the arrival and service processes used in sections 6.1.1 and 6.1.2 does not deliver the required theoretical mean number in the system for the temporal sequence of time intervals.

To compensate for this state of affairs the data stream of system entities must be manipulated by means of a designer equation(Appendix B) The designer equation is a necessary adjunct to equations (6.3) and (6.4) to shape the data stream of system entities to reflect reality of system operation modelled via passing reference to interevent times (arrival and service). The generation of the system state with the passage of time is driven by random number generation and is shown in Fig. 6.1.3. The adequacy of the generation process, which includes the use of a designer equation, is demonstrated by achieved results. The model can now be used in spreadsheet form for the analysis of steady state and transient operation of an M/M/1 queue. Consequently it may also serve as a touchstone in evaluating the use of the Chaos based models which follow. One should however not lose sight of the fact that the Poisson/exponential assumption is a mathematical concept and that no real process can be expected to constantly be in agreement with it. It is however heartening to know that use of it as a benchmark will lead to a conservative evaluation of alternative modelling methods.

**The Verhulst generated single channel queue**

If as at the outset of this chapter considering the use of chaos generation methods to model a single channel queueing system by means of approximations and numerical techniques is heeded, and robustness and simplicity of modelling is to be achieved, the concept of studying a temporal sequence of equal time intervals which accommodate arrival and service events is justified. As in the case of the classical M/M/1 queue analysis of par. 6.1.3.1 the Verhulst system model makes use of the highly simplified model described in equation (6.3) which also requires manipulation of the generated data stream by designer equations. The generation of the system state with the passage of time is driven by chaos iterative generation and is shown in Fig 6.2.3. The adequacy of the generation process, which includes the use of a designer equation, is demonstrated by the achieved results.

**Benchmarking the Verhulst generated single channel queue model**

Comparison of the Poisson M/M/1 and Verhulst methods of generating system dynamics as depicted in Figs. 6.1.3 and 6.2.3 respectively results in • achieving equivalence of mean and standard deviation values for the arrival and service processes, • achieving graphical plausibility of system orbit likeness i.e. applying the TLAR criterion (“that looks about right”) in comparing the two system entity orbits. No quantitative justification for “Poissonness” other than the foregoing parameter determination and application of the TLAR plausibility criterion has been carried out. As a further matter of interest the Verhulst methods of generating system dynamics over 200 one minute intervals are shown in Fig. 6.2.4 for a general service distribution queueing system for λ = 8,µ = 10 and σ = 0.010 . The average number of entities in the system is given .

**TABLE OF CONTENTS :**

**CHAPTER INTRODUCTION**- 1.1 INTRODUCTION
- 1.1.1 A General Description of the proposed Research
- 1.1.2 Exploring novel approaches to the modelling of Congestion

- 1.2 LITERATURE STUDY
- 1.2.1 Queues
- 1.2.1.1 Description of Queues
- 1.2.1.1 Historical perspective
- 1.2.1.3 Review of Queueing Models and their Modelling Approaches
- 1.2.1.4 Confidence limits

- 1.2.2 Chaos theory

- 1.2.1 Queues
- 1.2.2.1 Historical Perspective
- 1.2.2.2 Modelling Approach
- 1.2.3 The need for a new theory

- 1.1 INTRODUCTION
**CHAPTER CONFIDENCE LIMITS FOR EXPECTED WAITING TIME OF TWO QUEUEING MODELS**- 2.1.1 Introduction
- 2.1.3 The ML and CAN estimators for expected waiting time

- 2.1.3.1 The ML Estimator
- 2.1.3.2 An application of the multivariate central limit theorem
- 2.1.3.3 The CAN Estimator
- 2.1.4 Confidence limits for the expected waiting time
- 2.2 STATISTICAL ANALYSIS FOR A TANDEM QUEUE WITH BLOCKING
- 2.2.1 Introduction
- 2.2.2 System description and assumptions
- 2.2.3 Analysis of the system

- 2.2.3.1 Transient Solution
- 2.2.3.2 The Steady state solution
- 2.2.3.3 Expected service time per entity in the system
- 2.2.4 MLE and CAN estimator for the expected service time per entity in the
- system
- 2.2.4.1 The ML estimator
- 2.2.4.2 An application of the multivariate central limit theorem
- 2.2.4.3 The CAN Estimator
- 2.2.4.4 Confidence limits for the expected waiting time

**CHAPTER A SINGLE CHANNEL QUEUEING MODEL WITH OPTIONAL SERVICE AND SERVICE INTERRUPTION**- 3.1 Introduction
- 3.2 Model description
- 3.3 Assumptions and notation
- 3.4 Time dependant solution
- 3.5 Some special Cases
- 3.6 The steady state solution
- 3.7 Some special Cases
- 3.8 Concluding remark

**CHAPTER AN M/M/1 QUEUEING SYSTEM WITH BATCH ARRIVALS OF VARYING SIZE, SERVICE OF FIXED BATCH SIZE AND TWO MODES OF FAILURE OF SERVICE FACILITY**- 4.1 Introduction
- 4.2 Model description
- 4.3 Assumptions and notation
- 4.4 Equations describing the system
- 4.5 Time dependant solution
- 4.6 The steady state solution
- 4.7 Some special cases
- 4.8 Concluding remark

**CHAPTER AN M/G/1 QUEUEING SYSTEM WITH TWO MODES OF FAILURE**- 5.1 Introduction
- 5.2 Model description
- 5.3 System description
- 5.4 Equations governing the system
- 5.5 Time dependant solution
- 5.6 Steady state solution
- 5.7 Some special cases
- 5.8 Concluding remarks

**CHAPTER CHAOS THEORY BASED MODELS OF SIMPLE SYSTEMS OF CONGESTION**- 6.1 INTRODUCTION
- 6.1.1 The classical Poisson arrival system
- 6.1.1.1 The general modelling approach
- 6.1.2 The classical exponential service system
- 6.1.2.1 The general modelling approach

- 6.1.3 The classical M/M/1 queue
- 6.1.3.1 The general modelling approach
- 6.2 INTRODUCTION TO CHAOS GENERATION
- 6.2.1 The Verhulst generated arrival system
- 6.2.1.1 The general modelling approach

- 6.2. The Verhulst generated service system
- 6.2.2.1 The general modelling approach
- 6.2.3 The Verhulst generated single channel queue
- 6.2.3.1 The general modelling approach
- 6.2.4 Benchmarking the Verhulst generated single channel queue model

- 6.2.5 Extending the Verhulst generated single channel queue model to deal with variable traffic intensity
- 6.3 FURTHER EXAMPLES OF CHAOS GENERATION
- 6.4 CONCLUDING REMARKS ON SINGLE CHANNEL ORBITS RESULTING FROM A MENU OF METHODS OF GENERATION

- 6.1 INTRODUCTION
**CHAPTER ANALYSIS OF THE DYNAMIC CHARACTERISTICS OF PRACTICAL SYSTEMS OF CONGESTION USING CHAOS GENERATION METHODS**- 7.1 INTRODUCTION
- 7.2 SYSTEM NO
- 7.2.1 System scenario
- 7.2.2 The system model
- 7.2.3 Diagnosis of the model results
- 7.2.4 Using realtime feedback to improve system performance
- 7.2.5 The effect of the size of system waiting area on system performance
- 7.2.6 Concluding comments on system no

- 7.3 SYSTEM NO
- 7.3.1 System scenario
- 7.3.2 The system model
- 7.3.3 Diagnosis of the model results
- 7.3.4 Using realtime feedback to improve system performance
- 7.3.6 Concluding comments on system no

- 7.4 SYSTEM NO
- 7.4.1 System scenario
- 7.4.2 The system model
- 7.4.3 Diagnosis of the model results
- 7.4.4 Using realtime feedback to improve system performance
- 7.4.5 Concluding comments on system no

- 7.5 SYSTEM NO
- 7.5.1 System scenario
- 7.5.2 The system model
- 7.5.3 Diagnosis of the model results
- 7.5.4 Using realtime feedback to improve system performance
- 7.5.5 Concluding comments on System No

- 7.6 EVALUATION OF THE MODELLING METHODS AND ACHIEVEMENT OF DYNAMIC OPERATION RESULTS OF COMPLEX SYSTEMS OF CONGESTION

**CHAPTER CONCLUSION**- APPENDIX A
- FLOW DIAGRAM FOR THE DESIGN OF AN ARRIVALS/SERVICE ORBIT GENERATING FUNCTION
- APPENDIX B
- FLOW DIAGRAM FOR THE DESIGN OF A SYSTEM ORBIT GENERATING FUNCTION
- REFERENCES

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Stochastic models of steady state and dynamic operation of systems of congestion

Stochastic models of steady state and dynamic operation of systems of congestion