# Discrete-Event Simulation

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3 3 SINGLE-SERVER QUEUE One of the classic examples of discrete-event simulation is a Single-Server Queue. are two additional important assumptions. First, service is non-preemptive i.e. after initiation, a job will be continued to be serviced until completion. No other job can preempt the current job being serviced. Second, service is conservative i.e. the server will never sit idle if there is one or more jobs in the queue. Fig. 4. Possible job actions upon arrival Fig. 3. Representation of a Single-Server Queue In a Single-Server Queue, the Calling Population is infinite; that is, if a unit leaves the Calling Population and joins the waiting line or enters service, there is no change in the arrival rate of other units that could need service. Arrivals for service occur one at a time in a random fashion. Service times are some random length according to a probability distribution which does not change over time. The system capacity has no limit, meaning that any number of units can wait in line. Finally, units are served in the order of their arrival by a single server. Jobs (customers) arrive at the single server at random points in time in seek of service. When service is provided, the service time involved is also random. At the completion of service, jobs depart. The server operates as follows: as each new job arrives, if the server is busy then the job enters the queue, else the job immediately enters service; as each old job departs, if the queue is empty then the server becomes idle, else a job is selected from the queue to immediately enter service. At any time, the state of the server will either be busy or idle and state of the queue will be either empty or not empty. The Single-Server Queue model is built with the intention of answering the following relevant questions: What is the mean waiting time? What is the mean queue length? What is the mean length of a busy period? How does the performance change if we speed up the server? Control of the queue is determined by the queue discipline i.e. the algorithm used when a job is selected from the queue to enter service. The standard algorithms are: FIFO : first in, first out LIFO : last in, first out SIRO : service in random order Priority : typically, shortest job first (SJF) Certainly, the most common queue discipline is FIFO which is also known as FCFS (first come, first served). FIFO ensures that the order of arrival at the server and the order of departure from the server are the same. This observation leads to the simplification of the simulation model. There Fig. 5. Server outcomes after the completion of service 4 SPECIFICATION MODEL After their arrival to the service queue, jobs are indexed by i=1,2,3, For each job there are six associated time variables: The arrival time of job i is a i The delay of job i in the queue is d i 0. The time that job i begins service is b i = a i + d i. The service time of job i is s i > 0. The wait of job i in the service system (queue and service) is w i = d i + s i. The time that job i completes service (the departure time) is c i = a i + w i. The interarrival time between jobs i-1 and i is r i = a i a i-1. Fig. 6. Six time variables associated with job i. 4.1 ALGORITHMIC QUESTION Given a knowledge of the arrival times a 1, a 2,, the associated service times s 1, s 2,, and the queue discipline, how can the delay times d 1, d 2, be computed? If the queue discipline is FIFO then there is a simple algorithm for computing d i for all i. The delay d i of job i = 1, 2, 3, is determined by when the job s arrival time a i occurs relative to the departure time c i-1 of the previous job. There are two cases to consider. Case I. If a i < c i-1, i.e., if job i arrives 138

4 before job i-1 departs then job i will experience a delay of di = c i-1 a i. and average service time is: Fig. 7. Job i arrives before job i-1 departs. The reciprocal of the average interarrival time is the arrival rate and the reciprocal of the average service time is the service rate. Similarly, for the first n jobs, The average delay in the queue is: Case II. If instead a i c i-1, i.e., if job i arrives after (or just as) job i-1 departs then job i will experience no delay so that d i = 0. and the average wait for the service is: Fig. 8. Job i arrives after job i-1 departs So, it is clear that the truth of the expression a i < c i-1 determines whether or not job i will experience a delay ALGORITHM If the arrival times a 1, a 2, and service times s 1, s 2, are known and if the server is initially idle, then this algorithm computes the delays d 1, d 2, in a single-server FIFO service with infinite capacity. The three statistics w, d and s are together called as jobaveraged statistics i.e. the data is averaged over all jobs. Time-Averaged Statistics: three additional variables are required to define the time-averaged statistics for a singleserver queue : l(t) = 0, 1, 2, is the number of jobs in the whole system at time t; q(t) = 0, 1, 2, is the number of jobs in the queue at time t; x(t) = 0, 1 is the number of jobs in service at time t. By definition, l(t) = q(t) + x(t) for any t > 0. Over the time interval (0, T) the time-averaged number in the whole system is: and the time-averaged number in the queue is: OUTPUT STATISTICS One basic question that needs to be aptly answered is what statistics should be generated when constructing a discreteevent simulation. From a job s (customer s) perspective the most important statistic might be the average delay (the smaller the better), whereas from management s point of view, the server s utilization (the proportion of busy time) is the most important (the larger the better). Job-Averaged Statistics: for the first n jobs, The average interarrival time is: and the time-averaged number in service is: The three statistics l, q and x are together called as timeaveraged statistics. 5 ADVANTAGES OF SIMULATION a) Simulation allows the study of and experimentation with a complex system. b) It enables the feasibility testing of any hypothesis about how or why certain phenomena occur. 139

5 c) Flexibility in time handling as it can be compressed or expanded to allow for a speed-up or slow-down of the phenomena under investigation. d) Designing of a simulation model helps in gaining knowledge that could lead to the improvement of the system. e) Evaluating the different circumstances of simulation by changing the inputs and observing the resultant outputs can produce valuable insight into which variables are the most important. f) New hardware designs, physical layouts, transportation systems, and so on can be tested without committing resources for their acquisition. g) Simulation helps in formulation and verification of analytical solutions. much more faster tomorrow than it is today due to the advances being made in hardware everyday permitting rapid running of scenarios. Thus, in near future, very likely discrete-event simulation is going to prove itself ubiquitous with more efficient working and accurate results. REFERENCES [1] Jerry Banks, John S. Carson II, Barry L. Nelson and David M. Nicol, Discrete-Event System Simulation, 2011 [2] Lawrence Leemis and Steve Park, Discrete- Event Simulation: A First Course, DISADVANTAGES OF SIMULATION a) Special training is required to build simulation models. Very often, attempt to gain insights through simulation turns futile due to ambiguous models. b) Since so much randomness is associated with simulation (random inputs) so it can be hard to distinguish whether an observation is a result of system interrelationships or of randomness. c) Simulation modeling and analysis can be time consuming and expensive. 7 APPLICATIONS OF SIMULATION Computer Network Simulation: simulate new protocols for different network traffic scenarios before deployment. Business Process Simulation: agent-based modeling and simulation of store performance for personalized pricing, modeling and simulation of a telephone call center, human fatigue risk simulations in continuous operations. Hospital Applications: modeling front office and patient care in ambulatory health care practices, estimating maximum capacity in an emergency room, reducing the length of stay in an emergency department. Design Implementation: to overcome implementation problems occurring in typical program evaluations like attrition problems, data coding errors, floor and ceiling effects on measures etc. that degrade the theoretical quality of these designs. Vehicle Manufacturing: to study the outcome of various factors which affect the production process like absence of labor force, undermined manufacturing times, equipment failures, lack of work or blockage etc., provide animation possibility suitable for layout design. Transportation Modes and Traffic: simulating aircraft-delay absorption, runaway schedule determination by simulation optimization, simulation of freeway merging and diverging behavior etc. 8 CONCLUSION This paper has reviewed the flexibility provided by discreteevent simulation which can be used to model a wide variety of problems. Although it significantly facilitates the representation of complex systems but still there are a range of issues along the model development, parameter estimation, implementation and analysis that should be resolved to maximize the efficiency of the final output. Also, simulation should be avoided when the problem could be solved analytically because simulation can consume a lot of time and money. There is no doubt that simulations will be 140

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