MGT 181: Queueing Theory

• Queueing theory is a Management Science topic covered in MGT 181
• Asst. Prof. Vincent Crist Lipayo, MBM, PhD prepared the material

Objectives of Queueing Theory

• Define queueing theory, queues, queueing models, and queueing systems
• Describe the benefits of implementing a queueing system in various service settings
• Compare and contrast classes of queueing systems and types of queueing models
• Identify and categorize queueing systems in commercial, internal, and transportation services by system, customer, and server type
• Recognize and apply key constructs and formulas
• Solve problems using standard queueing models like single-server (M/M/1) and two-server (M/M/2) queues

Queue, Queueing Models, and Queueing Systems

• Queue: A waiting line for customers/items with limited resources/capacity
  • Examples: Bank teller lines, cars at toll booths, chat support users
• Queueing Model: Analytical framework representing a queue, with assumptions about arrival, service, and behavior
  • Examples: M/M/1 single-server, M/M/s multi-server, M/G/1 general service
• Queueing System: Real-world service environment
  • Includes customers, servers, and queue handling rules like first-come, first-served (FCFS)
  • Examples: Airport check-in, fast-food drive-thrus, hospital emergency rooms

Queueing System Common Benefits

• Helps determine the probability of no customers in the system
  • Highlights the importance of system capacity for higher utilization
  • Enables productive employers/servers leading to higher profit
• Determines the average number of customers in the queue
  • Helps with staffing decisions
• Determines the average time each customer spends in the queue to minimise customer waiting time
• Determines probability of customer waiting, emphasizing operational efficiency with enough servers
• Goal: Efficient operations

Classes of Queueing System

• Commercial Service Systems are used when outside customers receive service from commercial organizations
  • Examples: Bank tellers, fast-food counters, call centers, supermarket checkouts
  • Also includes retail fitting rooms, ticket counters (cinemas/events), online customer support chat, hospitals, pharmacies and hotel front desks.
• Internal Service Systems are used when customers are internal to the organization
  • Examples: IT helpdesk, internal print/copy centers, in-house maintenance requests
  • Also includes HR query counters, internal logistics/delivery, internal cafeteria lines, software testing queues
  • Further examples include internal approval workflows, finance payroll queries, and internal training registration.
• Transportation Service Systems pertain to vehicles being the customers
  • Examples: Airport check-in, bus terminals, and toll booths
  • Also includes subway turnstiles, ferry boarding lines, taxi stands, ride-hailing app queues, and parking garage entry queues

Key Constructs in Queueing Models

• Arrival Rate (λ): Average number of customers arriving per time unit
  • Example value: 10 customers/hour
• Service Rate (μ): Average number of customers served per time unit
  • Example value: 12 customers/hour
• Traffic Intensity (ρ): Ratio of arrival rate to service rate (system utilization)
  • Example value: 0.83
• Number of Servers (s): Total servers serving customers
  • Example value: 1 (single-server)
• Queue Length (Lq): Average number of customers waiting in the queue
  • Example value: 3.47 customers
• System Length (L): Average number of customers in the system (waiting + being served)
  • Example value: 4.3 customers
• Waiting Time in Queue (Wq): Average time customer spends waiting in the queue
  • Example value: 0.347 hours (≈21 minutes)
• Time in System (W): Average time customer spends in the system (waiting + service)
  • Example value: 0.43 hours (≈26 minutes)
• Queue Discipline: Rule for determining order of service (e.g., FCFS)
• System Capacity (K): Maximum customers allowed (e.g., ∞ for infinite)
• Calling Population: Source of customers (finite or infinite)

Formulas

• Traffic Intensity: ρ = λ / μ - Measures the system utilization and should be less than 1 for stability
• Average # in System (L): L = λ × W - Little's Law: total customers = arrival rate × time in system
• Average # in Queue (Lq): Lq = ρ² / (1 − ρ) - Expected number waiting (for M/M/1)
• Average Time in System (W): W = 1 / (μ − λ) - Total time spent in queue + service
• Average Waiting Time in Queue (Wq): Wq = λ / [μ(μ − λ)] - Average time customer waits before being served
• L from Lq and ρ: L = Lq + ρ - Total in system is the waiting time plus being served
• W from Wq and service rate: W = Wq + (1 / μ) - Total time in system is waiting time plus service time

Assumptions of Queueing Theory

• Interarrival times are independent and identically distributed according to distribution specified
• All arriving customers enter the queueing system and remain until service is complete
• Queueing system has a single, infinite-capacity queue
  • Can hold unlimited customers for all practical purposes
• Queue discipline is first come, first served (FCFS)
• Queueing system has specified number of servers, each capable of serving customers.
• Each customer is served individually by available servers
• Service times are independent and identically distributed according

Queueing Models

• Single-Server Queue
  • Notation: M/M/1
  • Arrival Process: Poisson
  • Service Time: Exponential
  • Servers: 1
  • Key Assumptions: Infinite queue, FCFS discipline, infinite population
  • Best Application Areas: Small service desks, single ATMs, basic customer support
• Two-Server Queue
  • Notation: M/M/2
  • Arrival Process: Poisson
  • Service Time: Exponential
  • Servers: 2
  • Key Assumptions: Same as M/M/1, but with 2 parallel servers
  • Best Application Areas: Bank counters and hospital reception desks
• Multi-Server Queue:
  • Notation: M/M/s
  • Arrival Process: Poisson
  • Service Time: Exponential
  • Servers: s (≥1) -Key Assumptions: General multi-server system
  • Best Application Areas: Call centers, airport check-ins, ticketing systems
• Deterministic Service:
  • Notation: M/D/1
  • Arrival Process: Poisson
  • Service Time: Deterministic
  • Servers: 1
  • Key Assumptions: Constant service time
  • Best Application Areas: Vending machines and car washes
• General Service Time
  • Notation: M/G/1
  • Arrival Process: Poisson
  • Service Time: General
  • Servers: 1
  • Key Assumptions: Flexible service distribution
  • Best Application Areas: Repair shops and medical appointments
• Deterministic Arrival
  • Notation: D/M/1
  • Arrival Process: Deterministic
  • Service Time: Exponential
  • Servers: 1
  • Key Assumptions: Fixed inter-arrival time
  • Best Application Areas: Scheduled maintenance tasks
• General Arrival & Service
  • Notation: G/G/1
  • Arrival Process: General
  • Service Time: General
  • Servers: 1
  • Key Assumptions: Fully generalized and often requires simulation
  • Best Application Areas: Complex and custom service systems in highly variable environments
• Finite Capacity Queue
  • Notation: M/M/1/K
  • Arrival Process: Poisson
  • Service Time: Exponential
  • Servers: 1
  • Key Assumptions: System can hold a maximum of K customers and those in excess are lost or rejected
  • Best Application Areas: Limited-space systems that include parking lots, elevators, and streaming buffers

Solving Single-Server Queues

• Prepare the data by tabulating the following in Excel - Arrival Rate (λ) - Service Rate (μ) - Confirm it's a Single Server (M/M/1) queue.
• Compute the Utilization (ρ) - By using: ρ = λ / μ - Check if ρ < 1 to ensure the system is stable.
• Calculate the Average Time in System (W) - By using W = 1 / (μ - λ) - Convert the time to minutes if needed.
• Calculate the Average Number in the System (L) - By using L=λ×W
• Calculate the Average Number in the Queue (Lq) - By using Lq = ρ² / (1 - ρ)
• Calculate the Average Waiting Time in Queue (Wq) - Wq = Lq / λ or Wq = λ / (μ*(μ - λ)) - Convert to minutes if needed.

Notation Reference Table

• Caret symbol (^) is used for exponentiation
• Parentheses () are used to group operation in the PEMDAS rule
• Square brackets [] are used for manual math notation and aren't recognized in Excel formulas
• Braces {} are used in set notation or programming and aren't functional in Excel formulas

Solving Server Queues (M/M/1) questions

• Sari-sari store example is provided that uses the single server queue model
• Laundry service example is provided that uses single server queue model

Comparing M/M/1 and M/M/2

• Queue:
  • Formula for finding arrival rate (ρ); for M/M/1 (ρ = λ /μ) and M/M/2 (ρ = λ / (s × μ)); not similar due to no. of ‘s’
  • Formula for finding average number (L); for M/M/1 (L = Lq + (λ / μ)) and M/M/2 (L = Lq + (λ / μ)); similar
  • Formula for finding queue length (Lq); for M/M/1 (Lq = ρ² / (1 - ρ)) M/M/2 (Lq = (Pw × λ × μ) / (s – λ/μ)²), not similar and M/M/2 is complex
  • Formula for finding the average customer wait time (W); for M/M/1 (W=L/λ) and M/M/2 (W = Wq + 1 / μ), not similar
  • Formula for finding queue waiting time (Wq); for M/M/1 (Wq = λ / (μ × (μ - λ))) M/M/2 (Wq = Lq / λ), only similar for single sever
  • Formula for finding queue waiting time (Wq) for both M/M/1 (Wq = Lq / λ) and M/M/2 (Wq = Lq / λ), similar
  • Formula for finding probability (Po) for both M/M/1(P₀ = [∑(λ/μ)^n / n! from n=0 to s−1 + (λ/μ)^s / (s!·(1−ρ))]⁻¹) and M/M/2(P₀ = [∑(λ/μ)^n / n! from n=0 to s−1 + (λ/μ)^s / (s! (1−ρ))]⁻¹), not applicable for M/M/1

Double-Server Queues

• Identify known values λ, μ, and s
• Find the traffic intensity ρ = λ / (s × μ)
• Compute for the P₀ (probability that zero customers ) use below formula
  • P₀ = [∑(λ/μ)^n / n! from n=0 to s−1 + (λ/μ)^s / (s!·(1−ρ))]⁻¹
• Calculate average number in queue
  • Lq = (Pw × λ × μ) / (s – λ/μ)²
• Calculate average number in system
  • L = Lq + (λ / μ)
• Calculate average time in system
  • W = L / λ
• Calculate average waiting time in queue
  • Wq = Lq / λ
• Results in terms of customers and time (convert hours to minutes if needed to summarise

Solving questions for Two-Server Queues (M/M/2)

• Sari-sari store (Keenan's Checkout Counter) example is provided that uses the Double Server Queue model

References

• This presentation uses content, information and data from many sources
  • Hillier, M., & Hillier, F. (2011). Introduction to management Science: a modeling and case studies approach with spreadsheets (International ed.). McGraw-Hill. Chapter 11.
  • Cinco, J. Jr. (2021). Video lecture 181.5.1 on queueing theory. UP Cebu, School of Management.
  • Cinco, J. Jr. (2021). Video lectures 181.5.2 on queueing theory. UP Cebu, School of Management.

Material coverage for Major Examinations

• LE2 (Long Exam 2) covers 50 objective and calculation based questions
• It includes content and questions from
  • Queueing theory
    • M/M/1
    • M/M/2 - some