Queuing Theory Overview

Queues: Queues are ordered sets of people, items, or tasks awaiting service.

• Queues are established based on a prioritization system according to the server's policies and procedures.

Queuing Systems: Describe the organization's structure for handling queues.

• These systems involve procedures, work stations, and personnel designed to accommodate and satisfy demand.

Why are there waiting lines?

• Underloaded System: When the server (person or machine) is overcapacity and cannot process customers at the expected rate.
• Irregularity of Arrivals: When customer arrivals are inconsistent and the server cannot maintain a steady pace.
• Large Demand: When the number of customers exceeds the server's ability to handle them.

Queuing Theory: Is a mathematical approach to analyzing waiting lines.

• It seeks to optimize system performance by minimizing cost and wait times while maximizing customer satisfaction and profit.

Population Source: Represents the potential number of people or items entering the queuing system.

• Finite Population Source: The maximum number of potential customers is known, representing a limited population.
• Infinite Population Source: The population source is either unlimited or indeterminate.

Number of Servers: Refers to the number of service channels (people or workstations) available to serve customers.

• Single Channel, Single Phase: One service point with a single step in the service process.
• Single Channel, Multiple Phases: One service point with multiple steps.
• Multiple Channels, Single Phase: Multiple service points with a single step.
• Multiple Channels, Multiple Phases: Multiple service points with multiple steps.

Arrival Patterns: Describe the regularity or irregularity of customer arrival intervals.

• Regular Arrivals: Customers arrive at predictable intervals.
• Irregular Arrivals: Customer arrivals happen unpredictably.

Service Patterns: Describe the server's method and time spent serving customers:

• Exponential Service Time: The server is a human, and service times vary.
• Constant Service Time: The server is a machine and provides services at a consistent rate.

Queue Discipline: Determines the order in which customers are served.

• First In, First Out (FIFO): The first customer in line is served first.
• First In, Last Out (FILO): The last customer arriving in line is served first.
• No Particular Order: Customers are served without adhering to a specific order.

Measures of System Performance: Evaluate the efficiency and effectiveness of a waiting line system.

• Average Number of Customers Waiting: Measures the average number of customers in the queue.
• Average Time Spent Waiting: Represents the average time customers spend waiting in line and being served.
• System Utilization: Measures the percentage of time the server is busy serving customers.
• Cost of Waiting: The cost of waiting for service, including the cost of lost productivity or customer dissatisfaction.
• Probability of Waiting: The likelihood that a customer will have to wait for service.

Symbols and Representation: Use symbols to represent various aspects of a queuing system.

• λ (lambda) - Customer arrival rate
• µ (mu) - Service rate
• 1/µ - Service time
• г (gamma) - Average number of customers being served at a given time
• ρ (rho)- System Utilization
• M - Number of servers
• Lq - Average number of customers waiting for service
• Ls - Average number of customers in the system
• Wq - Average time spent waiting in line
• Ws - Average time spent in the system

Basic Relationships: Formulas to calculate key performance metrics.

• System Utilization*: ρ = λ / (Mµ)
• Average Number of Customers Being Served*: г = λ / µ
• Average Number of Customers:*
• Waiting in line for service (Single Channel, Exponential Service Time)
  • Lq = λ² / (µ(µ - λ))
• Waiting in line for service (Single Channel, Constant Service Time)
  • Lq = λ² / (2µ(µ - λ))
• In the system:
  • Ls = Lq + г
• Average Time of Customers:*
• Waiting in line:
  • Wq = Lq / λ
• In the System:
  • Ws = Wq + 1/µ
  • Ws = Ls / λ
• Percent of Idle Time*:
• Percent of Idle Time = 1 - ρ
• Idle Time per hour = (1 - ρ) x 60 minutes