Index-Based Priority Rule

• Index-Based Priority Rule is a system that assigns numerical indices based on urgency, cost, and operational criteria to optimize resource allocation and scheduling.
• It employs lexicographic scoring and iterative algorithms to ensure high-priority tasks are served before lower-priority ones, maintaining order in complex systems.
• Widely used in queueing networks, combinatorial matchings, and restless bandit models, the rule supports decentralized decision-making and efficiency through strategic bidding and pricing.

An index-based priority rule is a class of policies for allocating resources, scheduling queues, or organizing combinatorial optimization steps based on numerical indices assigned to jobs, vertices, bandits, or customers. These indices embody urgency, cost-efficiency, or other operational criteria, and priority is enforced by lexicographic or ordering-based selection. Index-based priority rules form the foundation for optimal policies in canonical settings such as queueing games with time-accumulating priorities, combinatorial matchings with prioritized vertices, and stochastic control of restless bandit systems. These methodologies unify static and dynamic prioritization via indexed, state- or structure-dependent rules, supporting both decentralized and centralized optimization objectives across diverse stochastic and deterministic domains.

1. Priority Indices and Lexicographic Scoring

Index-based rules require the assignment of an explicit priority index, $\rho(u)$, to each object (vertex, job, or customer). For instance, in combinatorial matchings, $\rho(u)\in\{1,\dots,n\}$, with lower values corresponding to higher priority. The "priority score" of a structure $M$ (e.g., a matching) is defined as the $n$-ary integer

$S(M) = \sum_{i=1}^n d_i(M)\,n^{n-i},$

where $d_i(M)$ is the number of priority-$i$ vertices served by $M$. The maximum-priority rule seeks $M^*$ with lexicographically maximal $(d_1,\ldots,d_n)$. This scoring system supports hierarchical optimization: higher-priority counts are maximized before considering lower-priority improvements, exactly reflecting the structure of many real-world scheduling and resource allocation constraints (Turner, 2015).

For dynamic or stochastic systems such as queues and restless bandits, the index may be a function of time, state, or strategic choice. In an accumulating-priority M/G/1 queue, each customer selects a slope $b_i\geq0$, and her (time-evolving) priority index at time $t$ after arrival is $P_i(t) = b_i(t-s)$, with $s$ the arrival time (Haviv et al., 2015).

2. Algorithmic Frameworks for Maximizing Indexed Objectives

Index-based rules drive algorithmic methods that prioritize improvements at higher indices and maintain lexicographic invariants. In matching, this is realized via a modified Edmonds augmenting-path algorithm:

• The procedure handles priorities iteratively from highest to lowest.
• At each level $i$, searches for "i-augmenting paths" that increase $d_i$ without reduction in $d_1,\ldots,d_{i-1}$.
• Augmentation is performed until no such paths remain, ensuring lex-maximality of $(d_1,\ldots,d_n)$.
• Invariants enforce that priority improvements at level $i$ never degrade the previously maximized counts at higher priorities.

This rule enables an $O(mn)$ algorithm for achieving maximum-priority matchings, where $m=|E|$ and $n$ is the number of priorities. The general methodology extends to any optimization structure where high-priority entities must be served before lower-priority ones and is robust under extensions involving odd-length cycles ("blossoms") and contraction (Turner, 2015).

3. Strategic Bidding in Accumulating-Priority Queues

Haviv and Ravner present a queueing model in which customers purchase a linear "priority slope" $b$ so that their place in the queue, and thus their likelihood of prompt service, accumulates over time in proportion to $b$. At any time the server selects the customer with the greatest $P_i(t)=b_i(t-s)$, effecting an index-based selection that continuously depends on waiting time and user-determined index (Haviv et al., 2015).

Customers bid for $b$ to optimize their net cost $c_i(b; b_{-i}) = C_i W(b; b_{-i}) + b$, where $C_i$ is the per-unit waiting cost and $W(b;\,b_{-i})$ the expected waiting time given the bid vector. The Nash equilibrium profile of slopes $(b_1^e,\ldots,b_N^e)$ solves a first-order optimality condition incorporating the queue's stochastic structure and the interplay of priority indices.

• In homogeneous settings (equal $C$ and $G$), a unique equilibrium exists, and the rule reduces to FCFS.
• In heterogeneous settings, equilibrium slopes strictly increase with $C_i$, and the priority order $b_1^e < \cdots < b_N^e$ determines the effective service hierarchy.

This priority index regime supports both "follow-the-crowd" and "avoid-the-crowd" behaviors, depending on parameter ranges and the distribution of rival priorities.

4. Priority Policies and Whittle's Index in Restless Bandit Models

Index-based priority rules are fundamental to the solution of multi-class restless bandit problems and dynamic resource allocation. The fluid-equilibrium linear program for restless bandits identifies strict-priority schedules that order bandit classes by computed indices, such as the celebrated Whittle index, derived via Lagrange multipliers in a relaxation of the hard resource constraint (Verloop, 2016).

• Each bandit (indexed by state $(j, k)$) is assigned an index $\nu^f_k(j)$, indicating the minimal subsidy for which passivity is optimal.
• The system activates the $\alpha$ bandits with highest index at each time, realizing an index-based priority policy.
• Under indexability and a global attractor condition for the fluid ODEs, the resulting rule is provably asymptotically optimal.
• Even when indexability fails, robust asymptotically optimal policies can be synthesized by examining LP basic solutions and selecting priority orderings accordingly.

A worked example in the M/M/S+M queue yields an explicit index formula,

$$\iota_k = (\mu_k + \tilde{\theta}_k)\left( \frac{c_k + d_k \theta_k}{\theta_k} - \frac{\tilde{c}_k + \tilde{d}_k \tilde{\theta}_k}{\mu_k + \tilde{\theta}_k} \right),$$

and the resulting index rule aligns with Whittle's policy under suitable monotonicity (Verloop, 2016).

5. Pricing Mechanisms and Efficiency in Index-Based Priority Games

Index-based priority rules enable market-based or game-theoretic allocation mechanisms by linking the purchase of index (priority) to costs or prices. In the accumulating-priority queue, charging a price $p(x, b) = x b$ proportional to realized service time and slope $b$ modifies customers' effective cost to $C_i W(b; b^e) + \bar{x}_i b$ and realigns Nash equilibrium bids to match the social optimum C$\mu$-rule ordering, where priorities reflect $C_i/\bar{x}_i$ rather than $C_i$ alone (Haviv et al., 2015).

Numerical results demonstrate that without such pricing, decentralized equilibria may substantially exceed the optimum in total waiting cost. Properly calibrated, the price restores strong efficiency properties, shrinking the efficiency gap to within 10–20% in heavily loaded regimes. The rent dissipation phenomenon—where all like-priority customers converge to identical bids—occurs universally across these index-based bidding games.

6. Applications and Broader Context

Index-based priority rules are widely useful in:

• Queueing networks with time-dependent or strategic priorities.
• Combinatorial optimization, especially prioritized matching and allocation.
• Dynamic resource allocation in telecommunications, healthcare operations, and cloud computing.
• Restless bandit and multi-armed bandit problems, particularly under high-load or asymptotic scaling limits.

The index-based paradigm supports decentralized implementation and strong performance benchmarks, with tractable algorithms and provable optimality or near-optimality in varied stochastic, strategic, and combinatorial environments (Haviv et al., 2015, Turner, 2015, Verloop, 2016).