LINE Solver

The MAM solver analyzes open and closed queueing systems with Markovian structure using matrix-analytic methods. The core solver is built on top of the BuTools library [1], Q-MAM, SMCSolver, and MAMSolver [3]. Native algorithms for parametric decomposition of queueing networks are integrated with this solver.

Methods

The method names and defaults below describe the MATLAB interface. Aliases share a row; model-specific restrictions and backend differences are noted. Use solver.listValidMethods() to inspect the names available for your model.

Configuration options

The solver-specific fields below belong to options.config. Set them on an options struct, for example opt.config.name = value, and pass that struct to the solver constructor. See shared solver options for all top-level fields, shared configuration, defaults and usage. Options apply only to the methods and model features that consume them.

Relevant top-level options: iter_max, cutoff, timespan.

The MATLAB iteration limit is iter_max=100. Shared merge, compress and fixed-point driver settings also control decomposition. For INAP methods, use AG.

MAP/BMAP/1 Queue Solver

The MAM solver includes specialized support for MAP/BMAP/1 queues (Markovian Arrival Process with Batch Markovian Service). This solver models the queue as a GI/M/1 type Markov chain (skip-free to the right) and uses the ETAQA algorithm for efficient computation of stationary probabilities and queue length moments.

Key Characteristics:

• MAP arrivals: Arrivals follow a Markovian Arrival Process, adding one customer at a time
• BMAP service: Service completions can process batches of 1, 2, 3, ... customers
• GI/M/1-type structure: Level increases by exactly 1 (arrivals), decreases by any amount (batch service)

Features:

• Supports phase-type modulated arrival processes (MAP)
• Handles batch service with arbitrary batch size distributions
• Computes mean queue length, utilization, response time, and throughput
• Returns aggregated stationary probabilities and R matrix

MATLAB Usage:

% Create 2-phase MAP for arrivals
C0 = [-3.0, 1.0; 0.5, -2.0];  % transitions without arrivals
C1 = [1.5, 0.5; 0.5, 1.0];    % arrival transitions

% Create BMAP for service with batch sizes 1 and 2
D0 = [-5.0, 1.0; 0.5, -4.5];  % transitions without service
D1 = [2.0, 0.5; 0.5, 2.0];    % batch size 1 service
D2 = [1.0, 0.5; 0.5, 1.0];    % batch size 2 service

% Solve MAP/BMAP/1
[QN, UN, RN, TN] = solver_mam_map_bmap_1({C0, C1}, {D0, D1, D2});

% Display results
fprintf('Mean queue length: %.4f\n', QN);
fprintf('Utilization: %.4f\n', UN);
fprintf('Mean response time: %.4f\n', RN);
fprintf('Throughput: %.4f\n', TN);

Python Usage:

import numpy as np
from line_solver.api_native.mam import solver_mam_map_bmap_1

# Create 2-phase MAP for arrivals
C0 = np.array([[-3.0, 1.0], [0.5, -2.0]])
C1 = np.array([[1.5, 0.5], [0.5, 1.0]])

# Create BMAP for service with batch sizes 1 and 2
D0 = np.array([[-5.0, 1.0], [0.5, -4.5]])
D1 = np.array([[2.0, 0.5], [0.5, 2.0]])
D2 = np.array([[1.0, 0.5], [0.5, 1.0]])

# Solve MAP/BMAP/1
result = solver_mam_map_bmap_1(C0, C1, [D0, D1, D2])

print(f"Mean queue length: {result.mean_queue_length:.4f}")
print(f"Utilization: {result.utilization:.4f}")
print(f"Mean response time: {result.mean_response_time:.4f}")
print(f"Throughput: {result.throughput:.4f}")

BMAP/MAP/1 Queue Solver

The MAM solver includes specialized support for BMAP/MAP/1 queues (Batch Markovian Arrivals with Markovian Service). This solver models the queue as an M/G/1 type Markov chain (skip-free to the left) and uses the ETAQA algorithm for efficient computation of stationary probabilities and queue length moments.

Key Characteristics:

• BMAP arrivals: Batch arrivals with sizes 1, 2, 3, ... following a Batch Markovian Arrival Process
• MAP service: Service completions follow a Markovian Arrival Process, serving one customer at a time
• M/G/1-type structure: Level increases by any amount (batch arrivals), decreases by exactly 1 (single service)

Features:

• Supports batch arrivals with arbitrary batch size distributions
• Handles phase-type modulated service processes (MAP)
• Computes mean queue length, utilization, response time, and throughput
• Returns aggregated stationary probabilities and G matrix

MATLAB Usage:

% Create BMAP for arrivals with batch sizes 1 and 2
D0 = [-2.8, 1.6; 1.6, -2.8];  % transitions without arrivals
D1 = [0.5, 0.2; 0.2, 0.5];    % batch size 1 arrivals
D2 = [0.3, 0.2; 0.2, 0.3];    % batch size 2 arrivals

% Create 2-phase MAP for service
S0 = [-5.0, 1.0; 0.5, -4.5];  % transitions without service
S1 = [2.5, 1.5; 1.5, 2.5];    % service transitions

% Solve BMAP/MAP/1
[QN, UN, RN, TN] = solver_mam_bmap_map_1({D0, D1, D2}, {S0, S1});

% Display results
fprintf('Mean queue length: %.4f\n', QN);
fprintf('Utilization: %.4f\n', UN);
fprintf('Mean response time: %.4f\n', RN);
fprintf('Throughput: %.4f\n', TN);

Python Usage:

import numpy as np
from line_solver.api_native.mam import solver_mam_bmap_map_1

# Create BMAP for arrivals with batch sizes 1 and 2
D0 = np.array([[-2.8, 1.6], [1.6, -2.8]])
D1 = np.array([[0.5, 0.2], [0.2, 0.5]])
D2 = np.array([[0.3, 0.2], [0.2, 0.3]])

# Create 2-phase MAP for service
S0 = np.array([[-5.0, 1.0], [0.5, -4.5]])
S1 = np.array([[2.5, 1.5], [1.5, 2.5]])

# Solve BMAP/MAP/1
result = solver_mam_bmap_map_1([D0, D1, D2], S0, S1)

print(f"Mean queue length: {result.mean_queue_length:.4f}")
print(f"Utilization: {result.utilization:.4f}")
print(f"Mean response time: {result.mean_response_time:.4f}")
print(f"Throughput: {result.throughput:.4f}")