Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions Mcom sem 2 Delhi University
Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions MCOM sem 2 Delhi University:- we will provide complete details of Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions MCOM sem 2 Delhi University in this article.
Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions Mcom sem 2 Delhi University
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.
Queueing theory has its origins in research by Agner Krarup Erlang when he created models to describe the Copenhagen telephone exchange.The ideas have since seen applications including telecommunication, traffic engineering, computing and, particularly in industrial engineering, in the design of factories, shops, offices and hospitals, as well as in project management.
Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions Mcom sem 2 Delhi University:-Single queueing nodes
Single queueing nodes are usually described using Kendall’s notation in the form A/S/C where A describes the time between arrivals to the queue, S the size of jobs and C the number of servers at the node. Many theorems in queueing theory can be proved by reducing queues to mathematical systems known as Markov chains, first described by Andrey Markov in his 1906 paper.
Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on what would now be called queueing theory in 1909. He modeled the number of telephone calls arriving at an exchange by a Poisson process and solved the M/D/1 queue in 1917 and M/D/k queueing model in 1920. In Kendall’s notation:
- M stands for Markov or memoryless and means arrivals occur according to a Poisson process
- D stands for deterministic and means jobs arriving at the queue require a fixed amount of service
- k describes the number of servers at the queueing node (k = 1, 2,…). If there are more jobs at the node than there are servers then jobs will queue and wait for service
The M/M/1 queue is a simple model where a single server serves jobs that arrive according to a Poisson process and have exponentially distributed service requirements. In an M/G/1 queue the G stands for general and indicates an arbitrary probability distribution. The M/G/1 model was solved by Felix Pollaczek in 1930, a solution later recast in probabilistic terms by Aleksandr Khinchin and now known as the Pollaczek–Khinchine formula. ^{}^{}
After the 1940s queueing theory became an area of research interest to mathematicians. In 1953 David George Kendall solved the GI/M/k queue and introduced the modern notation for queues, now known as Kendall’s notation. In 1957 Pollaczek studied the GI/G/1 using an integral equation. John Kingman gave a formula for the mean waiting time in a G/G/1 queue: Kingman’s formula. ^{}
The matrix geometric method and matrix analytic methods have allowed queues with phase-type distributed inter-arrival and service time distributions to be considered.
Problems such as performance metrics for the M/G/k queue remain an open problem.^{}
Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions Mcom sem 2 Delhi University:-Purpose
- Simulation is often used in the analysis of queueing models
- Queueing models provide the analyst with a powerful tool for designing and evaluating the performance of queueing systems.
- Typical measures of system performance:
- Server utilization, length of waiting lines, and delays of customers
- For relatively simple systems, compute mathematically
- For realistic models of complex systems, simulation is usually required.
Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions Mcom sem 2 Delhi University:-Key elements of queueing systems
- Key elements of queueing systems
- Customer: refers to anything that arrives at a facility and requires service, e.g., people, machines, trucks, emails.
- Server: refers to any resource that provides the requested service, e.g., repairpersons, retrieval machines, runways at airport.
Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions Mcom sem 2 Delhi University:-Calling Population
- Calling population: the population of potential customers, may be assumed to be finite or infinite.
- Finite population model: if arrival rate depends on the number of customers being served and waiting, e.g., model of one corporate jet, if it is being repaired, the repair arrival rate becomes zero.
- Infinite population model: if arrival rate is not affected by the number of customers being served and waiting, e.g., systems with large population of potential customers.
Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions Mcom sem 2 Delhi University:-System Capacity
- System Capacity: a limit on the number of customers that may be in the waiting line or system.
- Limited capacity, e.g., an automatic car wash only has room for 10 cars to wait in line to enter the mechanism.
- Unlimited capacity, e.g., concert ticket sales with no limit on the number of people allowed to wait to purchase tickets
Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions Mcom sem 2 Delhi University:-Arrival Processes – Infinite population models
- In terms of interarrival times of successive customers.
- Random arrivals: interarrival times usually characterized by a probability distribution.
- Most important model: Poisson arrival process (with rate λ), where An represents the interarrival time between customer n − 1 and customer n, and is exponentially distributed (with mean 1/λ).
- Scheduled arrivals: interarrival times can be constant or constant plus or minus a small random amount to represent early or late arrivals.
- e.g., patients to a physician or scheduled airline flight arrivals to an airport.
- At least one customer is assumed to always be present, so the server is never idle, e.g., sufficient raw material for a machine.
Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions Mcom sem 2 Delhi University:-Arrival Processes – Finite population models
- Customer is pending when the customer is outside the queueing system, e.g., machine-repair problem: a machine is “pending” when it is operating, it becomes “not pending” the instant it demands service form the repairman.
- Runtime of a customer is the length of time from departure from the queueing system until that customer’s next arrival to the queue, e.g., machine-repair problem, machines are customers and a runtime is time to failure.
Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions Mcom sem 2 Delhi University:-Queue Behavior and Queue Discipline
- Queue behavior: the actions of customers while in a queue waiting for service to begin, for example:
- Balk: leave when they see that the line is too long,
- Renege: leave after being in the line when it’s moving too slowly,
- Jockey: move from one line to a shorter line.
- Queue discipline: the logical ordering of customers in a queue that determines which customer is chosen for service when a server becomes free, for example:
- First-in-first-out (FIFO)
- Last-in-first-out (LIFO)
- Service in random order (SIRO)
- Shortest processing time first (SPT)
- Service according to priority (PR).
Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions Mcom sem 2 Delhi University:-Service Times and Service Mechanism
- Service times of successive arrivals are denoted by S1, S2, S3.
- May be constant or random.
- {S1, S2, S3, . . . } is usually characterized as a sequence of independent and identically distributed random variables, e.g., exponential, Weibull, gamma, lognormal, and truncated normal distribution.
- A queueing system consists of a number of service centers and interconnected queues.
- Each service center consists of some number of servers, c, working in parallel, upon getting to the head of the line, a customer takes the 1st available server.
Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions Mcom sem 2 Delhi University:-Queueing notation I
- A notation system for parallel server queues: A/B/c/N/K, (due to Kendall) where
- A represents the interarrival-time distribution,
- B represents the service-time distribution,
- c represents the number of parallel servers,
- N represents the system capacity,
- K represents the size of the calling population.
Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions Mcom sem 2 Delhi University
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