Queuing Systems: Models, Metrics & Real-World Applications

Core Concepts

What is a Queuing System?

A queuing system is a mathematical model of a real-world scenario where customers (jobs, people, data packets) arrive, wait in a queue, and receive service from one or more servers. Key components:

Real-world analogy: A Khalti payment queue at a bank ATM:

• Customers (you) arrive randomly.
• Queue forms if the server (ATM) is busy.
• Service time depends on transaction complexity (fixed or variable).
• Departure: you leave after payment confirmation.

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Key Definitions and Metrics

1. Arrival Process

Describes how customers enter the system. Common models:

• Poisson Process: Arrivals are random, independent, and occur at a constant average rate λ (customers/hour). Example: eSewa users logging in during peak hours (6–9 PM).
• Deterministic: Fixed time intervals (e.g., buses arriving every 15 minutes at a bus stop).

2. Service Discipline

Rules for selecting the next customer to serve:

3. Performance Metrics

Critical for evaluating system efficiency:

• λ (Lambda): Arrival rate (customers/unit time).
• μ (Mu): Service rate (customers/unit time per server).
• ρ (Rho): Utilization factor = λ/μ (must be < 1 to avoid infinite queues).
• L: Average number of customers in the system (queue + service).
• Lq: Average number of customers in the queue (waiting).
• W: Average time a customer spends in the system.
• Wq: Average waiting time in the queue.

Little’s Law (the golden rule): $$L=\lambda W\text{and}{L}_q=\lambda W_q$$ Example: If 100 customers arrive/hour (λ = 100) and average wait time is 5 minutes (W = 5/60 hours), then $L=100\times \frac{5}{60}=8.33$ customers in the system.

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Classic Queuing Models

1. M/M/1 Model

• M: Markovian (Poisson arrivals, exponential service times).
• 1: Single server.
• Assumptions:
  • Arrivals follow Poisson process (rate λ).
  • Service times are exponentially distributed (rate μ).
  • Infinite queue capacity (no customer is turned away).
  • FIFO discipline.

Key Formulas: $$\rho =\frac{\lambda }{\mu }(\rho <1\text{ for stability})$$ $$L=\frac{\rho }{1-\rho },L_q=\frac{{\rho }^2}{1-\rho }$$ $$W=\frac{1}{\mu -\lambda },W_q=\frac{\rho }{\mu -\lambda }$$

Worked Example: NTC Customer Service Call Center

• Given:
  • Calls arrive at 12/hour (λ = 12).
  • Agent handles 15 calls/hour (μ = 15).
  • Single agent (M/M/1).
• Calculate: $$\rho =\frac{12}{15}=0.8$$ $$L=\frac{0.8}{1-0.8}=4\text{ calls in the system}$$ $$W=\frac{1}{15-12}=0.333\text{ hours}=20\text{ minutes}$$ Interpretation: On average, a caller waits 20 minutes before their call is answered, and there are 4 calls (including the one being served) in the system.

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2. M/M/c Model

• c: Multiple servers (e.g., multiple ATMs at a bank).
• Key Formulas (for $\rho <1$): $$L=\frac{P_0(\rho {)}^c}{c!(1-\rho {)}^2}+\rho$$ where $P_0$ is the probability of an empty system: $$P_0={\left[\sum _{n=0}^{c-1}\frac{(\rho {)}^n}{n!}+\frac{(\rho {)}^c}{c!(1-\rho )}\right]}^{-1}$$

Worked Example: Daraz Order Fulfillment Center

• Given:
  • Orders arrive at 20/hour (λ = 20).
  • Each of 3 workers (c = 3) processes 10 orders/hour (μ = 10).
  • Poisson arrivals, exponential service.
• Calculate:
  1. Utilization per server: $$\rho =\frac{\lambda }{c\mu }=\frac{20}{3\times 10}=0.667$$
  2. Probability of empty system ($P_0$): $$P_0={\left[\sum _{n=0}^2\frac{(0.667{)}^n}{n!}+\frac{(0.667{)}^3}{3!(1-0.667)}\right]}^{-1}\approx 0.125$$
  3. Average orders in system ($L$): $$L=\frac{0.125\times (0.667{)}^3}{6\times (1-0.667{)}^2}+0.667\approx 2.33$$ Interpretation: On average, 2.33 orders are in the system (queued or being processed).

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In the Real World

1. eSewa Payment Queues:

   • Idea Used: M/M/c model for handling multiple payment requests.
   • How: During Diwali, eSewa’s servers act as "c" servers processing transactions (service rate μ). If λ (transactions/minute) exceeds $c\times \mu$, queues form, increasing $W_q$. eSewa uses load balancing (adding more servers) to keep $\rho$ low.

2. Pathao Driver Dispatch:

   • Idea Used: Priority-based queuing (SJF-like for nearby riders).
   • How: Pathao’s algorithm prioritizes riders closest to available drivers (shortest "service time" to match). This reduces $W_q$ for urgent trips.

3. Nepal Traffic at Kathmandu Ring Road:

   • Idea Used: M/M/1 with finite queue (traffic lights as servers).
   • How: Vehicles arrive at rate λ (cars/minute) at a signal. If the green light (service rate μ) is busy, cars queue. During peak hours (7–9 AM), $\rho$ approaches 1, causing gridlock. Solutions: add more lanes (increase μ) or optimize signal timing (reduce λ spikes).

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Validation and Limitations

When to Use Queuing Models?

Limitations:

• Assumptions: Real systems rarely fit Poisson/exponential perfectly (e.g., traffic has patterns).
• Complexity: Models like M/G/1 (general service time) require numerical methods.
• Human Behavior: Customers may abandon queues (e.g., Daraz shoppers leaving if $W_q$ > 5 minutes).

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Exam Tip

1. Memorize Little’s Law: Always relate $L$, $W$, and $\lambda$. Examiners love questions like: "If a bank has 20 customers on average and each spends 5 minutes, what’s the arrival rate?" Answer: $\lambda =L/W=20/(5/60)=240$ customers/hour.

2. Stability Check: Always ensure $\rho <1$. If $\rho \ge 1$, the queue grows infinitely—instant fail if you ignore this.

3. Real-World Mapping:

   • Single server: ATM, toll booth.
   • Multiple servers: Hospital wards, Daraz fulfillment centers.
   • Priority queues: ICU patients, VIP customers.

4. Graphical Questions:

   • Sketch a queueing system diagram (like the Mermaid graph above) to explain scenarios.
   • Plot $L$ vs. $\rho$ to show how queues explode as $\rho$ approaches 1.

5. Numerical Traps:

   • Watch units! Convert minutes to hours or vice versa.
   • For M/M/c, calculate $P_0$ step-by-step—examiners test this.

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