Multiline Queues: Models & Applications

• Multiline queues are layered multi-queue systems that model diverse settings in probability, queueing theory, concurrent data structures, and algebraic combinatorics.
• In queueing theory, these systems analyze multiple servers and cyclic or balancing policies to optimize metrics such as waiting times and maximum queue lengths.
• In concurrent and combinatorial contexts, MultiQueues and multiline queue models facilitate relaxed priority operations and encode steady states, linking to exclusion processes and Macdonald polynomial formulations.

Multiline queues denote several related but technically distinct constructions across probability, queueing theory, concurrent data structures, and algebraic combinatorics. In queueing theory, the term is used for systems with multiple queues or multiple servers, including parallel queues with simultaneous arrivals, polling systems, and multi-server queues whose extreme congestion behavior is analyzed asymptotically. In interacting-particle and combinatorial settings, multiline queues are arrays of occupied and vacant sites, or more general ball systems, that encode steady states of multispecies exclusion processes and give combinatorial formulas for Macdonald-type polynomials. In parallel algorithms, the closely related term MultiQueue refers to a relaxed concurrent priority queue built from multiple internal sequential priority queues (Badila et al., 2012, Finch, 2019, Corteel et al., 2018, Williams et al., 15 Apr 2025).

1. Terminological scope and principal meanings

Within the literature, “multiline queue” does not designate a single universal model. Instead, it labels several families of objects that share a multi-layer or multi-queue architecture.

Usage	Core object	Representative source
Queueing systems	Multiple physical or logical queues, servers, or lines	(Liu et al., 2013, Finch, 2019, Perel et al., 2022, Badila et al., 2012)
Concurrent data structures	Multiple internal sequential priority queues with relaxed semantics	(Williams et al., 2021, Williams et al., 15 Apr 2025, Postnikova et al., 2021)
Combinatorial probability	Layered arrays of occupied/vacant sites or balls encoding exclusion processes	(Ayyer et al., 2012, Corteel et al., 2018, Mandelshtam et al., 2024)

In the queueing-theoretic sense, the emphasis is on arrivals, service disciplines, workloads, waiting times, stability, and extremal congestion. In the combinatorial sense, multiline queues are discrete objects whose rows encode species, priorities, or queue sizes, and whose projections recover configurations of the multispecies TASEP, PASEP, or TAZRP on a ring (Ayyer et al., 2012, Pahuja, 2023, Mandelshtam et al., 2024). In the concurrent-algorithmic sense, MultiQueues are engineered relaxed priority queues whose semantics are quantified through rank error and delay rather than exact delete-min correctness (Williams et al., 2021, Williams et al., 15 Apr 2025).

This multiplicity of meanings is not merely terminological. It reflects a structural recurrence: layered systems in which local operations on rows or queues induce a tractable global law. A plausible implication is that the persistence of the term across fields is tied to this shared layered architecture rather than to a single common formalism.

2. Queueing-theoretic multiline systems

A central queueing instance is the service system with two independent queues receiving Poisson arrivals at rates $\lambda_1$ and $\lambda_2$, a single server, and size-independent service times: when the server serves a queue, all waiting customers in that queue are served in one fixed time unit, regardless of queue length (Liu et al., 2013). The state is $(x,y)$, where $x$ and $y$ are the numbers of waiting customers in queues 1 and 2, and the problem is formulated as an infinite-horizon Markov Decision Process with Bellman equation

$$V(x,y) = \lambda + \min\left\{ \gamma \mathbb{E}\big[V(Z_1, Z_2 + y)\big] + y,\; \gamma \mathbb{E}\big[V(Z_1 + x, Z_2)\big] + x \right\},$$

where $\lambda = (\lambda_1+\lambda_2)/2$, $Z_1,Z_2$ are Poisson arrivals, and $\gamma$ is the discount factor (Liu et al., 2013). The objective is to minimize the total expected discounted waiting time. The paper proves that among state-independent policies the optimal policy is cyclic: serve the slow-arrival queue once and the fast-arrival queue $k$ times, repeating. If $\lambda_2$0, then the optimal $\lambda_2$1 satisfies

$\lambda_2$2

with asymptotic behavior $\lambda_2$3 as $\lambda_2$4 and $\lambda_2$5 as $\lambda_2$6 (Liu et al., 2013).

A distinct multiline setting is the parallel-queue model with simultaneous arrivals. Here $\lambda_2$7 parallel $\lambda_2$8 queues each receive one job from the same Poisson arrival epoch, with service vector $\lambda_2$9 and ordered service times

$(x,y)$0

almost surely (Badila et al., 2012). With workloads $(x,y)$1, the recursion at arrival epochs is

$(x,y)$2

The stationary joint workload transform

$(x,y)$3

admits explicit formulas in the ordered case, together with a stochastic decomposition and a dual interpretation in multidimensional risk theory (Badila et al., 2012). In the two-dimensional general non-ordered case, the transform satisfies the kernel equation

$(x,y)$4

leading to a Riemann boundary value problem (Badila et al., 2012).

Multi-server queues also appear in extreme-value form. For the M/M/$(x,y)$5 queue, customers arrive according to a Poisson process at rate $(x,y)$6, each of $(x,y)$7 servers provides exponential service at rate $(x,y)$8, and $(x,y)$9 denotes the maximum queue length of idle customers over $x$0 (Finch, 2019). In the stable regime $x$1, the Poisson clumping heuristic yields

$x$2

with explicit specializations for $x$3 and a discrete Gumbel interpretation (Finch, 2019). The corresponding mean and variance approximations are

$x$4

$x$5

(Finch, 2019). The discrete-time Geo/Geo/2 analogue yields parallel formulas for maximum line length and, via a limiting argument with $x$6 and $x$7, recovers the M/M/2 case (Finch, 2019).

A recurring operational conclusion in these extreme-value analyses is the “one fast server is better than multiple slow servers” phenomenon for fixed total service capacity. In the emergency-room example with $x$8, one server with $x$9 yields smaller expected maximum queue length than two servers each with $y$0, and the average queue length comparison gives $y$1 for M/M/1 against $y$2 for M/M/2 (Finch, 2019). The M/M/$y$3 study reports the same qualitative pattern for maximum queue length constants when $y$4 is held fixed (Finch, 2019).

3. Polling, routing, and balancing policies for multiple queues

Multiline queueing models also include polling systems in which a single server attends several queues and routing interacts with service priorities. In the three-queue JSQ-SLQ polling system, arriving customers follow the Join the Shortest Queue discipline while the single server preemptively serves the longest queue (Perel et al., 2022). To avoid the intractable infinite three-dimensional state space $y$5, the process is represented as a two-dimensional Markov process with $y$6 and $y$7, where $y$8 is the queue being served and $y$9, $$V(x,y) = \lambda + \min\left\{ \gamma \mathbb{E}\big[V(Z_1, Z_2 + y)\big] + y,\; \gamma \mathbb{E}\big[V(Z_1 + x, Z_2)\big] + x \right\},$$0 (Perel et al., 2022).

Two complementary analytic frameworks are developed. The first is a Probability Generating Function system with 12 generating functions $$V(x,y) = \lambda + \min\left\{ \gamma \mathbb{E}\big[V(Z_1, Z_2 + y)\big] + y,\; \gamma \mathbb{E}\big[V(Z_1 + x, Z_2)\big] + x \right\},$$1 satisfying

$$V(x,y) = \lambda + \min\left\{ \gamma \mathbb{E}\big[V(Z_1, Z_2 + y)\big] + y,\; \gamma \mathbb{E}\big[V(Z_1 + x, Z_2)\big] + x \right\},$$2

where $$V(x,y) = \lambda + \min\left\{ \gamma \mathbb{E}\big[V(Z_1, Z_2 + y)\big] + y,\; \gamma \mathbb{E}\big[V(Z_1 + x, Z_2)\big] + x \right\},$$3 is a $$V(x,y) = \lambda + \min\left\{ \gamma \mathbb{E}\big[V(Z_1, Z_2 + y)\big] + y,\; \gamma \mathbb{E}\big[V(Z_1 + x, Z_2)\big] + x \right\},$$4 matrix. The second recasts the model as a Quasi-Birth-Death process with 12 phases and infinite levels, with steady-state vectors

$$V(x,y) = \lambda + \min\left\{ \gamma \mathbb{E}\big[V(Z_1, Z_2 + y)\big] + y,\; \gamma \mathbb{E}\big[V(Z_1 + x, Z_2)\big] + x \right\},$$5

and stability condition

$$V(x,y) = \lambda + \min\left\{ \gamma \mathbb{E}\big[V(Z_1, Z_2 + y)\big] + y,\; \gamma \mathbb{E}\big[V(Z_1 + x, Z_2)\big] + x \right\},$$6

(Perel et al., 2022).

The queue-balancing effect of the combined policy is quantified by the Gini Index

$$V(x,y) = \lambda + \min\left\{ \gamma \mathbb{E}\big[V(Z_1, Z_2 + y)\big] + y,\; \gamma \mathbb{E}\big[V(Z_1 + x, Z_2)\big] + x \right\},$$7

The reported numerical values are very small, for example $$V(x,y) = \lambda + \min\left\{ \gamma \mathbb{E}\big[V(Z_1, Z_2 + y)\big] + y,\; \gamma \mathbb{E}\big[V(Z_1 + x, Z_2)\big] + x \right\},$$8, rising up to $$V(x,y) = \lambda + \min\left\{ \gamma \mathbb{E}\big[V(Z_1, Z_2 + y)\big] + y,\; \gamma \mathbb{E}\big[V(Z_1 + x, Z_2)\big] + x \right\},$$9 only in extreme asymmetric cases (Perel et al., 2022). The paper attributes this to the complementary balancing effects of JSQ, which sends arrivals to shorter queues, and SLQ, which allocates service to longer queues.

This balancing perspective is consistent with the two-queue size-independent service model, where the optimal state-independent schedule is also a regular cyclic schedule rather than a reactive one (Liu et al., 2013). Taken together, these results suggest that in some multi-queue environments, structured cyclic or balancing policies can approximate or realize optimal performance without requiring full state observability. Because this conclusion synthesizes results from different models, it is best read as an interpretation rather than as a theorem valid uniformly across queueing systems.

4. MultiQueues as relaxed concurrent priority queues

In parallel computing, MultiQueues are relaxed concurrent priority queues composed of multiple internal sequential priority queues. In the 2021 formulation, the data structure consists of an array $\lambda = (\lambda_1+\lambda_2)/2$0 of $\lambda = (\lambda_1+\lambda_2)/2$1 sequential priority queues, where $\lambda = (\lambda_1+\lambda_2)/2$2 is the number of parallel threads and $\lambda = (\lambda_1+\lambda_2)/2$3 is a tuning parameter (Williams et al., 2021). Insert operations choose a random unlocked queue; delete-min chooses two random queues and removes the minimum from the queue with the smaller observed minimum. The quality of this relaxation is measured by rank error and delay.

For the classical MultiQueue, the probability that an element of global rank $\lambda = (\lambda_1+\lambda_2)/2$4 is deleted next is

$\lambda = (\lambda_1+\lambda_2)/2$5

which implies

$\lambda = (\lambda_1+\lambda_2)/2$6

and with high probability the rank error is $\lambda = (\lambda_1+\lambda_2)/2$7 (Williams et al., 2021). The delay also has a geometric law with expected value $\lambda = (\lambda_1+\lambda_2)/2$8 (Williams et al., 2021). The architecture is augmented by insertion and deletion buffers, batch operations on the internal queues, and stickiness, where a thread preferentially reuses the same internal queue for several operations to improve cache locality (Williams et al., 2021).

The 2025 engineering study generalizes this design. The MultiQueue now uses an array of $\lambda = (\lambda_1+\lambda_2)/2$9 independent internal sequential priority queues, each protected by a lightweight lock, random insertions, and deletions that sample $Z_1,Z_2$0 queues and lock the one with the smallest observed minimum (Williams et al., 15 Apr 2025). The paper formalizes relaxed semantics with

$Z_1,Z_2$1

and

$Z_1,Z_2$2

so expected rank error scales as $Z_1,Z_2$3 and high-probability maximum rank errors as $Z_1,Z_2$4; delay follows the same geometric distribution (Williams et al., 15 Apr 2025). Buffering, batching, $Z_1,Z_2$5-ary heaps, and stickiness are reported as central throughput mechanisms, while “wait-free locking” is presented as the key locking discipline: because the number of queues exceeds the number of threads, operations complete in an expected constant number of lock attempts (Williams et al., 15 Apr 2025).

A scheduler-level variant is the Stealing Multi-Queue (SMQ), in which each thread has a local queue, steals with probability $Z_1,Z_2$6, and uses task batching (Postnikova et al., 2021). Under a fair-enough scheduler and sufficiently large $Z_1,Z_2$7, the paper states expected maximum-rank and average-rank bounds of the form

$Z_1,Z_2$8

and

$Z_1,Z_2$9

respectively, reducing to $\gamma$0 average rank and $\gamma$1 maximum rank when $\gamma$2, $\gamma$3, and $\gamma$4 is constant (Postnikova et al., 2021). The implementation is evaluated on graph-processing benchmarks, and the paper reports that it can surpass schedulers such as Galois and PMOD (Postnikova et al., 2021).

These MultiQueue results are not queueing models in the classical stochastic-service sense, but they retain the core multi-line motif: several internal queues are used to trade exact minimality for throughput, locality, and scalability. The terminology is therefore adjacent rather than identical to that of probabilistic multiline queues.

5. Combinatorial multiline queues and exclusion processes

In combinatorics and integrable probability, multiline queues are layered arrays that project to multispecies exclusion processes on a ring. In the inhomogeneous multispecies TASEP on a ring, a multiline queue is an $\gamma$5 cylindrical array of occupied and vacant sites, where row $\gamma$6 contains $\gamma$7 occupied sites (Ayyer et al., 2012). Ferrari–Martin ringing path transitions define a continuous-time Markov chain on these arrays, and the bully-path projection maps each multiline queue to a TASEP configuration. In the homogeneous case the stationary distribution on multiline queues is uniform, so the stationary probability of a TASEP configuration is proportional to the number of multiline queues projecting to it (Ayyer et al., 2012). For the inhomogeneous case, a conjectural monomial stationary weight is written in terms of coverage statistics $\gamma$8: $\gamma$9 with proved special cases for $k$0 and for a single first-class particle (Ayyer et al., 2012).

The multispecies ASEP on a circle leads to a weighted refinement. In the 2018 work linking multiline queues to Macdonald polynomials, a multiline queue is an $k$1 ball system with weakly increasing numbers of balls from top to bottom, and $k$2 denotes the set of multiline queues of type $k$3 (Corteel et al., 2018). Each nontrivial pairing receives a $k$4-weight, and the total generating function is

$k$5

The main results are

$k$6

with the ASEP stationary probability recovered from $k$7 (Corteel et al., 2018).

For the multispecies PASEP on a ring, the corresponding construction is the linked multiline queue with a $k$8-bully path algorithm (Pahuja, 2023). Rows contain prescribed numbers of occupied sites, and links between adjacent rows are weighted by $k$9 when the $\lambda_2$00th available target is chosen, where

$\lambda_2$01

The paper computes adjacent-particle correlations on the first two sites by reducing to projected 3-species PASEP configurations and summing weights over linked multiline queues (Pahuja, 2023).

The 2024 study of twisted multiline queues extends these constructions to arbitrary compositions rather than only partition shapes (Mandelshtam et al., 2024). A fermionic MLQ of type $\lambda_2$02 is a tuple $\lambda_2$03 with $\lambda_2$04 and $\lambda_2$05, whereas a bosonic MLQ replaces subsets by multisets (Mandelshtam et al., 2024). The fermionic twisted algorithm generalizes the Ferrari–Martin algorithm and is shown to be equivalent to the Arita–Ayyer–Mallick–Prolhac algorithm; the bosonic twisted algorithm is novel and generalizes the Kuniba–Maruyama–Okado construction for the TAZRP (Mandelshtam et al., 2024). A new Markov chain on bosonic twisted multiline queues is defined, projects to the $\lambda_2$06-TAZRP, and commutes with the symmetric-group action on rows (Mandelshtam et al., 2024).

Another probabilistic use of “multi-line process” occurs in the multiclass Hammersley–Aldous–Diaconis process. Here the invariant measures for the two-class process are constructed from a stationary M/M/1 queue with Poisson arrivals of rate $\lambda_2$07 and service attempts of rate $\lambda_2$08, placing first-class particles at departure times $\lambda_2$09 and second-class particles at unused service times $\lambda_2$10 (0707.4202). The construction generalizes to $\lambda_2$11 classes through tandem queues and yields a multi-line process whose invariant measure is a product of Poisson processes (0707.4202). This is not the same combinatorial MLQ as in TASEP, but it shares the queue-coupling and multilayer logic.

6. Spectral, crystal, and polynomial extensions

A major development is the embedding of multiline queues into the theory of symmetric functions and crystal combinatorics. In “Multiline queues with spectral parameters,” an MLQ of type $\lambda_2$12 is a sequence of queues $\lambda_2$13 where $\lambda_2$14 is a $\lambda_2$15-queue and acts on words by the generalized Ferrari–Martin algorithm (Aas et al., 2018). For a packed word $\lambda_2$16, the spectral weight is

$\lambda_2$17

summing over MLQs such that $\lambda_2$18, and for any permutation $\lambda_2$19 the $\lambda_2$20-twisted spectral weight satisfies

$\lambda_2$21

(Aas et al., 2018). This proves the commutativity conjecture of Arita, Ayyer, Mallick, and Prolhac. For a special family of words $\lambda_2$22, the spectral weight has the determinant formula

$\lambda_2$23

where $\lambda_2$24 is the complete homogeneous symmetric function (Aas et al., 2018).

At $\lambda_2$25, multiline queues yield formulas for $\lambda_2$26-Whittaker polynomials through a collapsing insertion procedure (Mandelshtam et al., 2024). For a multiline queue $\lambda_2$27, the major index $\lambda_2$28 gives

$\lambda_2$29

Collapsing is expressed through lowering operators

$\lambda_2$30

and is identified with crystal-operator dynamics (Mandelshtam et al., 2024). The same framework recovers Lascoux–Schützenberger charge formulas for $\lambda_2$31-Whittaker polynomials, Littlewood–Richardson coefficients, and the dual Cauchy identity (Mandelshtam et al., 2024).

The interpolation extension is given by signed multiline queues. In the 2025 work on interpolation Macdonald polynomials, a signed multiline queue is an enhanced $\lambda_2$32 ball system with classic rows and signed rows $\lambda_2$33, where balls in signed rows may carry positive or negative labels (Dali et al., 2 Oct 2025). The generating functions

$\lambda_2$34

satisfy

$\lambda_2$35

so interpolation Macdonald polynomials are obtained as generating functions of signed multiline queues (Dali et al., 2 Oct 2025). The construction reduces to the earlier multiline queue formula when no negative balls are present (Dali et al., 2 Oct 2025).

Across these algebraic developments, multiline queues function as a common combinatorial skeleton for steady states of exclusion processes, Macdonald polynomials, interpolation analogues, and crystal-theoretic symmetries. This suggests that the queueing interpretation is not merely metaphorical: it furnishes an organizing principle capable of transporting methods between integrable probability and symmetric-function theory.