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Statistics 2021 Paper II 50 marks Solve

Paper II — Q3

(a) Explain M|G|1 queuing system. Obtain Pollaczek-kinchine formula. (15 marks) (b) Use MODI method to solve the above…

(a)

Explain M|G|1 queuing system. Obtain Pollaczek-kinchine formula. 15 marks

(b)

Use MODI method to solve the above transportation problem:

Store I II III IV A 4 6 8 13 B 13 11 10 8 C 14 4 10 13 D 9 11 13 8

Supply 50 70 30 50

Demand 25 35 105 20 15 marks

(c)

Use two-phase method to solve: Maximize z = 2x₁ + x₂ + x₃ subject to the constraints 4x₁ + 6x₂ + 3x₃ ≤ 8 3x₁ - 6x₂ - 4x₃ ≤ 1 2x₁ + 3x₂ - 5x₃ ≥ 4 and x₁, x₂, x₃ ≥ 0. 20 marks

हिंदी में प्रश्न पढ़ें
(a)

पंक्ति प्रणाली M|G|1 की व्याख्या कीजिए । पोलेकजेक-किंचिन सूत्र को प्राप्त कीजिए । (15 अंक)

(b)

निम्नलिखित परिवहन समस्या का MODI विधि का उपयोग करके हल निकालिए :

भंडार I II III IV A 4 6 8 13 B 13 11 10 8 C 14 4 10 13 D 9 11 13 8

पूर्ति 50 70 30 50

मांग 25 35 105 20 (15 अंक)

(c)

द्विप्रावस्था विधि का उपयोग करके हल कीजिए : अधिकतमीकरण z = 2x₁ + x₂ + x₃ निम्न प्रतिबंधों के अंतर्गत 4x₁ + 6x₂ + 3x₃ ≤ 8 3x₁ - 6x₂ - 4x₃ ≤ 1 2x₁ + 3x₂ - 5x₃ ≥ 4 और x₁, x₂, x₃ ≥ 0. (20 अंक)

Q3 of the 2021 UPSC Mains Statistics Paper II, as printed
The question as printed in the 2021 Statistics paper
Model answer coming soon See all 2021 Statistics questions

The figure this question refers to, in words

The question paper is a scan and the diagram did not survive as text. This is the figure as read from the original page — every component, value and label — so the question can be worked from the text below.

(b) Table with rows labeled A, B, C, D and columns labeled I, II, III, IV, Supply. The values in the table are: Row A: 4, 6, 8, 13, 50 Row B: 13, 11, 10, 8, 70 Row C: 14, 4, 10, 13, 30 Row D: 9, 11, 13, 8, 50 Below the table, a row labeled Demand contains the values: 25, 35, 105, 20.

What "Solve" is asking you to do

Choose the method, then carry it through to a final answer. Identifying what kind of problem this is and why that method applies is the first thing marked; a correct figure arrived at invisibly earns almost nothing.

Structure that answers it

Given data and what is required → method chosen, with the reason it applies → set-up (equation, circuit, free body, trial balance) → working, step by step → answer with units and any condition of validity

Where marks are lost

Doing the middle steps mentally and writing only the result. In mathematics papers, a further loss comes from giving a decimal where the exact value in surds or fractions was wanted, or from skipping the justification a part explicitly asks for.

All UPSC directive words, compared →

How this answer will be evaluated

Approach

Framework: Operations Research (Queuing Theory, Transportation, Linear Programming). (a) explain: definition/context > points in order > small example > short close | (b) calculate: given > formula > substitution > result with units > interpretation | (c) calculate: given > formula > substitution > result with units > interpretation Full marks: Flawless derivations and calculations with clear, logical steps and correct final answers.

Key points expected

  • Define M/G/1 notation (Poisson arrivals, General service, 1 server)
  • State Little's Law (L = λW)
  • Derive mean queue length Lq using generating functions
  • State final formula Lq = (λ²σ² + ρ²) / 2(1-ρ)
  • Find Initial Basic Feasible Solution (e.g., Vogel's Approximation)
  • Calculate u_i and v_j potentials for basic cells
  • Compute net evaluations (Δ_ij) for non-basic cells
  • Iterate until all Δ_ij ≥ 0 to find optimal solution

Evaluation rubric

Each sub-part is marked on its own, against the marks and word limit printed on the paper.

  1. (a) Define M/G/1 system and derive the Pollaczek-Khinchine formula. 15 marks

    explain— definition/context → points in order → small example → short close

    Must cover

    • Define M/G/1 notation (Poisson arrivals, General service, 1 server)
    • State Little's Law (L = λW)
    • Derive mean queue length Lq using generating functions
    • State final formula Lq = (λ²σ² + ρ²) / 2(1-ρ)

    Loses marks

    • Confusing M/G/1 with M/M/1
    • Missing the variance term σ² in the final formula
    • Derivation without stating assumptions

    Earns more

    • Define traffic intensity ρ = λ/μ
    • Mention stability condition ρ < 1
    • Show step-by-step algebraic manipulation

    Extra mark

    • Mention M/M/1 as a special case (σ² = 1/μ²)
  2. (b) Solve the transportation problem using the MODI method. 15 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • Find Initial Basic Feasible Solution (e.g., Vogel's Approximation)
    • Calculate u_i and v_j potentials for basic cells
    • Compute net evaluations (Δ_ij) for non-basic cells
    • Iterate until all Δ_ij ≥ 0 to find optimal solution

    Loses marks

    • Arithmetic errors in cost calculation
    • Failing to check optimality condition (Δ_ij ≥ 0)
    • Incorrect assignment of u and v values

    Earns more

    • Correctly identify degenerate solution if applicable
    • Show clear table of u, v, and Δ values
    • Calculate final minimum transportation cost

    Extra mark

    • Check for alternative optimal solutions (Δ_ij = 0)
  3. (c) Solve the Linear Programming problem using the two-phase method. 20 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • Convert constraints to standard form (add slack/surplus/artificial variables)
    • Phase I: Minimize sum of artificial variables (W)
    • Phase II: Maximize original objective function (Z)
    • Perform simplex iterations until optimal solution is reached

    Loses marks

    • Incorrect setup of Phase I objective function
    • Arithmetic errors in simplex tableaus
    • Failing to remove artificial variables before Phase II

    Earns more

    • Correctly identify entering and leaving variables
    • Show clear simplex tableaus for both phases
    • State final values of x1, x2, x3 and Z

    Extra mark

    • Check for unboundedness or infeasibility explicitly

Model answer coming soon

Every evaluation on this site is marked against a verified model answer. This question's answer is still being written; evaluation opens the moment it lands.

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