Comparison of the Discriminatory Processor Sharing Policies

Discriminatory Processor Sharing policy introduced by Kleinrock is of a great interest in many application areas, including telecommunications, web applications and TCP flow modelling. Under the DPS policy the job priority is controlled by the vector of weights. Verifying the vector of weights it is possible to modify the service rates of the jobs and optimize system characteristics. In the present paper we present the results concerning the comparison of two DPS policies with different weight vectors. We show the monotonicity of the expected sojourn time of the system depending on the weight vector under certain condition on the system. Namely, the system has to consist of classes with means which are quite different from each other. The classes with similar means can be organized together and considered as one class, so the given restriction can be overcame.

💡 Research Summary

The paper investigates the performance of the Discriminatory Processor Sharing (DPS) scheduling discipline, focusing on how the choice of weight vectors influences the expected sojourn time of jobs in a multi‑class queueing system. DPS, originally introduced by Kleinrock, allocates service capacity among classes proportionally to a set of positive weights; this mechanism is relevant to TCP flow control, web server request handling, weighted round‑robin scheduling, and other resource‑sharing contexts.

The authors consider a system with M classes. Class k receives Poisson arrivals at rate λ_k and has exponentially distributed service requirements with mean 1/µ_k. The overall load ρ = Σ_k λ_k/µ_k is assumed to be less than one, guaranteeing stability. For a weight vector g = (g₁,…,g_M) > 0, each job in class k receives instantaneous service rate g_k / Σ_i g_i N_i, where N_i is the current number of jobs in class i. When all weights are equal, DPS reduces to the classic Processor Sharing (PS) discipline.

A central problem is to compare two weight vectors, α and β, and to determine which yields a smaller expected total sojourn time T_DPS(g) = Σ_k (λ_k/λ) T_k(g), where T_k(g) are class‑specific mean sojourn times. Prior work showed that T_k(g) can be obtained as the solution of a linear system of equations (Equation 1 in the paper). The authors rewrite this system in matrix form, introducing matrices A(g) and D(g) that capture cross‑class interactions, and define σ(g)_{ij} = g_j µ_i / (g_i µ_j + g_j µ_i).

The main theoretical contribution is Theorem 1. It states that if both α and β belong to the set G (i.e., their components are non‑increasing: α₁≥α₂≥…≥α_M and similarly for β) and if the ratios satisfy α_{i+1}/α_i ≤ β_{i+1}/β_i for all i = 1,…,M−1, then the expected sojourn time under α is no larger than under β, provided an additional condition on the service‑rate means holds: µ_{j+1}/µ_j ≤ 1−ρ for every j = 1,…,M−1. This condition ensures that the means of the classes are sufficiently separated; it is sufficient but not necessary, and it becomes less restrictive as the overall load ρ decreases.

The proof proceeds by expressing T_DPS(g) as a quadratic form involving the inverse of (E−B(g)), where B(g)=A(g)+D(g). The authors introduce the vector y = 1ᵗ (E−B(α))^{-1} M, with M = diag(µ₁,…,µ_M). Lemma 7 shows that if y’s components are ordered non‑increasingly (y₁≥y₂≥…≥y_M), then the difference T_DPS(α)−T_DPS(β) is non‑positive. Lemma 8 establishes that the ordering of y is guaranteed precisely under the mean‑separation condition µ_{j+1}/µ_j ≤ 1−ρ. Lemma 5 and Lemma 6 link the ratio condition on α and β to monotonicity properties of σ(·)_{ij}, which in turn affect the sign of the terms in the quadratic form.

When the mean‑separation condition is violated for a pair of adjacent classes, the authors propose setting the corresponding weights equal (α_{j+1}=α_j, β_{j+1}=β_j). This adjustment restores the required ordering of y and thus preserves the monotonicity result. Consequently, classes with similar means can be merged and assigned identical weights, effectively relaxing the restrictive condition.

The paper also extends the result to the case where arrival rates λ_i differ from one. By scaling all λ_i to a common factor q (or by representing rational λ_i as p_i/q and using continuity arguments for real λ_i), the same monotonicity theorem holds without altering the condition on the means.

Numerical experiments illustrate the theory. With three classes (M=3) and weight vectors parameterized as g(x) = (x−1, x−2, x−3) / Σ_i (x−i) for x>1, the authors plot T_DPS(g(x)) versus x. In the first scenario (µ₁=160, µ₂=14, µ₃=1, λ_i=1, ρ≈0.91) the mean‑separation condition is satisfied, and the plot shows a monotone decrease of T_DPS as x grows, confirming the theorem. In a second scenario (µ₁=5, µ₂=4, µ₃=3, λ_i=1, ρ≈0.9) the condition fails, and the curve exhibits non‑monotonic behavior, demonstrating the necessity of the condition.

Overall, the study provides a clear, mathematically rigorous guideline for weight selection in DPS systems: assign larger weights to classes with larger service rates, ensure that the ratios of successive weights are ordered consistently, and, when class means are close, treat those classes as a single group with equal weight. This guidance can be directly applied to design of fair and efficient resource‑allocation mechanisms in telecommunications, data centers, and operating‑system schedulers.

The authors conclude that the monotonicity result deepens the understanding of DPS dynamics, offers a practical sufficient condition for performance improvement, and opens avenues for further research on optimal weight design under more general service‑time distributions or in overloaded regimes.