Generalized Queue Replacement Principle (GQRP)

• GQRP is a framework that replaces true equilibrium queue delays with dual capacity shadow prices derived from a relaxed, LP-based formulation in dynamic user equilibrium.
• It sequentially solves two LPs—one for determining the cost pattern (including queueing delays) and one for flow patterns—ensuring exact equivalence under certain conditions.
• Validated on networks like Sioux Falls, GQRP offers computational efficiency and policy insights, while highlighting limitations for many-to-many OD models and heterogeneous users.

Generalized Queue Replacement Principle (GQRP) denotes, in the dynamic user equilibrium assignment problem with route and departure time choice, an equivalence between the equilibrium queueing delay pattern and the solution to a linear programming problem obtained by relaxing certain conditions in the original DUE-RDTC problem. In the many-to-one network setting, it provides a criterion under which the true equilibrium queues can be replaced by dual capacity shadow prices from a simpler LP, and it yields an exact dynamic user equilibrium by sequentially solving two LPs: one for the equilibrium cost pattern, including the queueing delay pattern, and one for the equilibrium flow pattern (Sakai et al., 10 Aug 2025).

1. Formal concept and scope

The setting is a many-to-one road network with $N+1$ nodes, where $N$ are origins and one is a single destination. Each directed link $(i,j)\in L$ has bottleneck capacity $\mu_{ij}$ and free-flow travel time $c_{ij}$. Each link consists of a bottleneck section followed by a free-flow section; when inflow exceeds capacity $\mu_{ij}$, a point queue forms at the bottleneck, and FIFO is assumed. Users are homogeneous: they share the same value of time, normalized to $\alpha=1$, the same preferred arrival time at destination $t^{\mathrm P}$, and the same schedule delay cost function $s(t)$, which is convex in $t$, satisfies $N$0, and obeys $N$1 for all $N$2 (Sakai et al., 10 Aug 2025).

A key modeling feature is a Lagrangian-like coordinate system in which variables are indexed by destination arrival time $N$3, not by departure time at origins. The principal state variables are $N$4, the destination-arrival rate of users whose origin is $N$5; $N$6, the destination-arrival rate of users who pass link $N$7; $N$8, the queueing delay on link $N$9 experienced by users who arrive at destination at time $(i,j)\in L$0; and $(i,j)\in L$1, the earliest travel time from node $(i,j)\in L$2 to the destination for users arriving at time $(i,j)\in L$3. In this coordinate system, the generalized cost of arriving at time $(i,j)\in L$4 using path $(i,j)\in L$5 is

$(i,j)\in L$6

The dynamic user equilibrium with route and departure time choice is defined as a state where each user chooses route and departure time so that their experienced travel cost is minimal, and no unused route–departure-time alternative offers a lower cost, given congestion caused by all users. Within this formulation, GQRP identifies conditions under which the equilibrium queueing delay pattern can be recovered from a queue-free, capacity-constrained optimization problem, rather than from the full queueing equilibrium model (Sakai et al., 10 Aug 2025).

2. DUE-RDTC formulation in Lagrangian coordinates

The DUE is formulated as a mixed linear complementarity problem with variables $(i,j)\in L$7, where $(i,j)\in L$8 is the constant equilibrium cost for origin $(i,j)\in L$9. Let $\mu_{ij}$0 be the node–link incidence matrix excluding the destination, $\mu_{ij}$1 the same matrix with negative entries replaced by $\mu_{ij}$2, and $\mu_{ij}$3 a diagonal matrix with diagonal entries $\mu_{ij}$4. The model is characterized by six conditions (Sakai et al., 10 Aug 2025).

Demand conservation requires

$\mu_{ij}$5

Elementwise,

$\mu_{ij}$6

Flow conservation at nodes requires

$\mu_{ij}$7

Queueing dynamics in Lagrangian coordinates are expressed by the point-queue complementarity condition

$\mu_{ij}$8

Elementwise, if $\mu_{ij}$9, then

$c_{ij}$0

whereas if $c_{ij}$1, then

$c_{ij}$2

The consistency condition

$c_{ij}$3

ensures Lipschitz continuity of cumulative arrivals and physical consistency.

The route-choice condition is

$c_{ij}$4

so that any link with positive flow lies on a minimum-cost path. The departure-time-choice condition is

$c_{ij}$5

which enforces that all used origin–arrival-time choices have equal and minimal cost $c_{ij}$6, and unused ones are weakly more expensive.

Collecting these conditions yields the problem denoted $c_{ij}$7. Using a standard LCP–QP equivalence, the model is also represented as a quadratic program $c_{ij}$8 with objective

$c_{ij}$9

where the differential term $\mu_{ij}$0 is the source of the main computational difficulty (Sakai et al., 10 Aug 2025).

3. Relaxed cost determination and the formal definition of GQRP

The generalized queue replacement principle is introduced by relaxing the original queueing and consistency conditions. Specifically, the consistency condition $\mu_{ij}$1 is removed, and the original queue dynamics are replaced by the simpler complementarity

$\mu_{ij}$2

Elementwise, if $\mu_{ij}$3, then $\mu_{ij}$4; if $\mu_{ij}$5, then $\mu_{ij}$6. This is an abstract capacity-constraint complementarity rather than a true queue-dynamics condition (Sakai et al., 10 Aug 2025).

Replacing the original queueing conditions by this relaxed relation produces the cost determination LCP, denoted $\mu_{ij}$7. Its equivalent optimization form, $\mu_{ij}$8, has objective

$\mu_{ij}$9

which is linear because the term $\alpha=1$0 disappears. The resulting model decomposes into a primal–dual LP pair.

The primal LP, $\alpha=1$1, is

$\alpha=1$2

subject to demand conservation, flow conservation, and

$\alpha=1$3

It can be interpreted as a capacity-constrained, queue-free flow problem.

The dual LP, $\alpha=1$4, is

$\alpha=1$5

subject to

$\alpha=1$6

and

$\alpha=1$7

In this dual problem, the variables $\alpha=1$8 behave like shadow prices of capacity constraints on links and are interpretable as queueing-delay candidates (Sakai et al., 10 Aug 2025).

If $\alpha=1$9 denotes the DUE solution and $t^{\mathrm P}$0 the solution of the cost determination problem, then GQRP holds when

$t^{\mathrm P}$1

This is the formal statement that the true equilibrium queues can be replaced by the dual capacity shadow prices of a simpler LP (Sakai et al., 10 Aug 2025).

4. Verification, exact solution, and sufficient conditions

The methodology for determining whether GQRP holds is two-stage. First, one solves $t^{\mathrm P}$2 and, optionally, $t^{\mathrm P}$3 to obtain a candidate equilibrium cost pattern $t^{\mathrm P}$4 and corresponding capacity-constrained flows. Second, one solves a flow determination LP, denoted $t^{\mathrm P}$5, that reinstates the stricter queueing structure while keeping the candidate cost pattern fixed (Sakai et al., 10 Aug 2025).

Given $t^{\mathrm P}$6, define

$t^{\mathrm P}$7

The flow determination LP is

$t^{\mathrm P}$8

subject to demand conservation,

$t^{\mathrm P}$9

and

$s(t)$0

Because $s(t)$1, and $s(t)$2 are constants in this step, $s(t)$3 is again an LP (Sakai et al., 10 Aug 2025).

The central characterization is exact: GQRP holds if and only if $s(t)$4 is feasible and its optimal value is zero. In that case,

$s(t)$5

Thus the LP-derived cost pattern is not merely approximate; it is the DUE cost pattern itself (Sakai et al., 10 Aug 2025).

The paper also gives a constructive sufficient condition in terms of the schedule delay slope. For any origin $s(t)$6 and times with positive demand flow $s(t)$7,

$s(t)$8

Under this condition, the flow pattern

$s(t)$9

with

$t$0

is feasible for $t$1 and yields zero objective value. The implication is that, when the slope condition holds, no second optimization is needed: equilibrium flows can be constructed analytically from the cost determination solution (Sakai et al., 10 Aug 2025).

A further policy-based result shows that GQRP can be enforced by scaling the late-arrival part of the schedule delay function. For

$t$2

there exists some $t$3 such that GQRP holds under $t$4 (Sakai et al., 10 Aug 2025).

5. Relation to the classical queue replacement principle

The immediate antecedent of GQRP is the Queue Replacement Principle (QRP) developed for corridor problems with heterogeneous commuters. In that setting, the network is a linear corridor with multiple bottlenecks, one destination, multiple origins, and commuters heterogeneous in the value of schedule delay. The DSO problem is solved analytically by combining a bottleneck-based decomposition property with optimal transport; the DUE solution is then obtained by proving that, at each bottleneck $t$5 and time $t$6, the optimal congestion price in the DSO state equals the queueing delay in the DUE state: $t$7 This equality is the formal statement of QRP in the corridor model (Sakai et al., 2022).

The corridor analysis gives a constructive sufficient condition on the slope of the schedule delay cost. If

$t$8

for every $t$9 and every $N$00, then QRP holds. A simplified sufficient condition independent of $N$01 is also given: $N$02 Under these conditions, the DUE solution is obtained by replacing prices in the DSO solution by queues while keeping the same arrival-time supports (Sakai et al., 2022).

GQRP generalizes this logic from single bottleneck or corridor settings to general many-to-one networks with route and departure time choice. The 2025 formulation explicitly states that it is a network-level generalization of QRP: the same equivalence between queueing delays and queue-eliminating prices is retained, but it is expressed in an arc-based LP framework built on the Lagrangian-like coordinate system rather than on a corridor decomposition alone (Sakai et al., 10 Aug 2025).

6. Queue-theoretic sample-path interpretation

A broader queue-theoretic interpretation emerges from the recursive representation of the $N$03 queue. That work does not mention the Generalized Queue Replacement Principle by name, but it gives a fully explicit, sample-path recursion for the arrival, service-completion, and departure epochs in a $N$04 queue. Because those recursions are built only from max, min, and addition, they are perfectly suited to support GQRP-type pathwise monotonicity and coupling arguments (Krivulin, 2012).

For a $N$05 queue with interarrival times $N$06, service times $N$07, arrival times $N$08, service-completion times $N$09, and departure times $N$10, the recursion is

$N$11

$N$12

and

$N$13

where $N$14 is the $N$15th smallest completion time among the first $N$16 customers. The ordering operator itself is represented using only $N$17, $N$18, and $N$19, and for $N$20 the departure recursion simplifies to

$N$21

For $N$22, the representation collapses to the classical $N$23 recursion (Krivulin, 2012).

The sample-path significance is monotonicity. In the recursive system, arrivals, completions, departures, and waiting times are deterministic functions of primitive inputs formed entirely by coordinatewise nondecreasing operations. This yields pathwise statements of the form: if service times are replaced by coordinatewise larger values, then completion times and departure times are coordinatewise larger; if interarrival times are replaced by coordinatewise smaller values, arrivals and departures occur earlier. A plausible implication is that the max/min/plus structure in queueing theory supplies an algebraic backbone for replacement principles in more general network settings, including the LP-based GQRP formulation (Krivulin, 2012).

7. Computational behavior, policy significance, and limitations

The computational appeal of GQRP is that, after discretization of time, all optimization problems are LPs. The first-stage flow LP is structurally a minimum-cost flow with capacity constraints, the dual cost LP is a sparse potential problem, and the second-stage $N$24 has the same order of magnitude as the first-stage primal problem. For the Sioux Falls and Eastern Massachusetts networks, the whole two-LP process runs in less than a second on a standard laptop, and the resulting $N$25 objective is $N$26, effectively exact (Sakai et al., 10 Aug 2025).

The paper reports four classes of numerical examples. In a small Braess network with $N$27 nodes, $N$28 links, and demands $N$29, $N$30, and $N$31, the sufficient condition holds and flows are constructed analytically. In the Sioux Falls network with $N$32 nodes, $N$33 links, and $N$34 origins, and in the Eastern Massachusetts network with $N$35 nodes, $N$36 links, and $N$37 origins, the equilibrium cost pattern $N$38 equals $N$39 exactly over the time intervals with positive $N$40, and the evaluated $N$41 objective is again $N$42. In the Nguyen network, two scenarios were studied: a GQRP-valid case with $N$43, where the $N$44 objective is approximately $N$45, and a no-GQRP case with $N$46, where the objective is approximately $N$47 and the violations are mostly in the queueing condition and are small and localized in time and space (Sakai et al., 10 Aug 2025).

The policy interpretation is equally explicit. Under the assumption that, in DSO, node-passing times are uniquely determined by destination arrival time, the KKT conditions of the DSO LP coincide with the cost determination optimality conditions. If GQRP holds, then

$N$48

where $N$49 is the DSO congestion price pattern. If the road manager implements dynamic pricing equal to the equilibrium queueing delay at all bottlenecks, every user’s generalized travel cost stays the same, the manager collects revenue equal to the eliminated queueing delay cost, and a Pareto improvement is achieved (Sakai et al., 10 Aug 2025).

The main limitations are stated directly. The framework is restricted to many-to-one networks; extension to many-to-many OD patterns is nontrivial. Demand is fixed rather than elastic. The queue model is a deterministic point queue with FIFO and does not directly model spillback. Users are homogeneous, and heterogeneous preferences require further work. GQRP does not always hold: if schedule delay costs are very steep, the LP-based queue pattern may be incompatible with true queue dynamics. Suggested extensions include heterogeneous users, stochastic demands or travel times, more general queueing models with spillback, fully many-to-many OD patterns, sharper necessary and sufficient conditions for GQRP, and the use of cost determination solutions as high-quality initial points for more general DUE algorithms when GQRP fails (Sakai et al., 10 Aug 2025).