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Questions and Answers
What are the three main components of a queueing system?
What are the three main components of a queueing system?
Which of the following is NOT a type of performance measure in queueing models?
Which of the following is NOT a type of performance measure in queueing models?
What does the 'm' in Kendall notation represent?
What does the 'm' in Kendall notation represent?
What would be included in the service time distribution of a queueing system?
What would be included in the service time distribution of a queueing system?
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Which parameter affects how many customers can wait in line in a queueing system?
Which parameter affects how many customers can wait in line in a queueing system?
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What is the primary purpose of queueing models?
What is the primary purpose of queueing models?
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What could be an application of queueing models?
What could be an application of queueing models?
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Which of the following describes service discipline in a queueing system?
Which of the following describes service discipline in a queueing system?
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What condition must be met for a queuing system to be stable?
What condition must be met for a queuing system to be stable?
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Which scenario describes a finite buffer system?
Which scenario describes a finite buffer system?
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How does Little's Law apply to the waiting facility specifically?
How does Little's Law apply to the waiting facility specifically?
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Given an I/O rate of 100 requests per second and an average response time of 0.1 seconds, what is the mean number of requests at the disk server?
Given an I/O rate of 100 requests per second and an average response time of 0.1 seconds, what is the mean number of requests at the disk server?
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Which statement about finite population systems is correct?
Which statement about finite population systems is correct?
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What does infinite buffer capacity imply in queuing models?
What does infinite buffer capacity imply in queuing models?
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Which distribution is most commonly associated with service time in queuing models?
Which distribution is most commonly associated with service time in queuing models?
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What does the symbol $λ$ represent in queuing theory?
What does the symbol $λ$ represent in queuing theory?
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Which of the following statements about interarrival times is true?
Which of the following statements about interarrival times is true?
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What does the notation G/G/1 indicate about a queuing system?
What does the notation G/G/1 indicate about a queuing system?
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Which variable is represented by $n$ in queuing theory?
Which variable is represented by $n$ in queuing theory?
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What is the key characteristic of a FCFS service discipline?
What is the key characteristic of a FCFS service discipline?
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What is the characteristic of a single server queue?
What is the characteristic of a single server queue?
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In queuing models, what type of arrival process is most commonly assumed?
In queuing models, what type of arrival process is most commonly assumed?
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Which service discipline serves customers based on their arrival order?
Which service discipline serves customers based on their arrival order?
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In a multiple server queue, which aspect does the number of servers impact the most?
In a multiple server queue, which aspect does the number of servers impact the most?
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What type of service discipline prioritizes customers with the shortest remaining service time?
What type of service discipline prioritizes customers with the shortest remaining service time?
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Which distribution is characterized by random and unpredictable arrival times?
Which distribution is characterized by random and unpredictable arrival times?
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What does the notation 'M/M/3/20/1500/FCFS' signify about the system?
What does the notation 'M/M/3/20/1500/FCFS' signify about the system?
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Which service discipline would serve customers based on their service time expectations?
Which service discipline would serve customers based on their service time expectations?
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What is a hyperexponential distribution primarily used for in queuing models?
What is a hyperexponential distribution primarily used for in queuing models?
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Study Notes
Introduction to Queuing Theory
- Queuing theory studies lines of waiting customers needing service.
- A queuing system is a waiting room, customers, and service providers.
- Queuing models estimate performance metrics.
- Examples include: waiting in a supermarket, waiting for information, and planes circling before landing.
Queuing Systems
- A queue is a line of waiting customers needing service from one or more providers.
- A queuing system includes customers, a waiting line, and service providers.
Queuing Models
- Queuing models are used to estimate system performance measures.
- Queuing models provide rough estimates for: server utilization, length of waiting lines, and delays of customers.
- Applications include determining the minimum number of servers, detecting bottlenecks, and evaluating system designs.
Introduction
- Waiting to pay in a supermarket is an example of a queuing system.
- Waiting at the telephone for information is a queuing system.
- Planes circling before landing is a queuing system.
Example Questions
- What is the average waiting time for a customer?
- How many customers are waiting on average?
- How long is the average service time?
- What is the chance that a server has nothing to do?
Queueing Notation
- Arrival Process, Service Time Distribution, Number of Servers, Buffer Capacity, Population Size, and Service Discipline.
Parameters of a Queuing System
- The behavior of a queuing system depends on: arrival process (and distribution of inter-arrival times), service process (and distribution of service times), number of servers, system capacity, and size of the population.
Kendall Notation
- A/S/m/B/K/SD (A: arrival process, S: service time distribution, m: number of servers, B: number of buffers (system capacity), K: population size, SD: service discipline)
Number of Servers
- Number of servers available
- Single server queue
- Multiple server queue
Service Disciplines
- First-come, first-served (FCFS)
- Last-come, first-served (LCFS)
- Shortest processing time first (SPT)
- Shortest remaining processing time first (SRPT)
- Shortest expected processing time first (SEPT)
- Shortest expected remaining processing time first (SERPT)
- Biggest-in-first-served (BIFS)
- Loudest-voice-first-served (LVFS)
Common Distributions
- Exponential
- Erlang with parameter k
- Hyperexponential with parameter k (mixture of k exponentials)
- Deterministic (constant)
- General (all)
Example
- M/M/3/20/1500/FCFS
- Time between successive arrivals is exponentially distributed.
- Service times are exponentially distributed.
- Three servers, 20 buffers, 1500 total jobs (20 buffers = 17 waiting + 3 service), jobs lost after 20.
- Service discipline is first-come-first served.
Defaults
- Infinite buffer capacity
- Infinite population size
- FCFS service discipline
- The first three of the six parameter are sufficient.
- Assume only individual arrivals
Arrival Process
- Arrival times (t₁, t₂, ..., tⱼ)
- Interarrival times (τⱼ = tⱼ - tⱼ₋₁)
- A sequence of independent and identically distributed (IID) random variables.
- Most common: Poisson arrivals (inter-arrival times are exponential and IID).
Service Time Distribution
- Amount of time each customer spends.
- Assume IID (independent and identically distributed).
- Most commonly used: exponential distribution (M, E, H, or G)
Key Variables
- Arrival rate (λ): mean arrival rate = 1/ E[τ]
- Inter-arrival time (τ): time between successive arrivals
- Service rate (μ): mean service rate per server = 1/ E [s]
- Total service rate for m servers: mμ
- Service time (s): Service time per job
- Number of jobs in the queue (n₁): Number of jobs waiting,
- Number of jobs receiving service(n₂):
- Number of jobs in the system (n): Number of jobs in the waiting line and the service line, also called queue length
- Response time (r): time in the system = waiting time + service time
- Waiting time (w): time between arrival and beginning of service
Stability Condition
- For stability, the mean arrival rate should be less than the mean service rate (λ < mµ).
Little's Law
- Mean number in the system = Arrival rate × Mean response time
Application of Little's Law
- Applied to the waiting facility of a service center: Mean number in the queue = Arrival rate × Mean waiting time.
- Applied to the service facility: Mean number in service = Arrival rate × Mean service time
Example using Little's Law
- Average time to satisfy an I/O request was 100 milliseconds.
- I/O rate was 100 requests per second.
- Mean number of requests in the disk server= 10.
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Description
This quiz explores the fundamental concepts of queuing theory, which examines lines of waiting customers and the systems that serve them. It covers queuing systems, models, and their applications in real-world scenarios like supermarkets and air travel. Test your understanding of performance metrics and system design.
