MEC Queue Digital Twin

The MEC Queue Digital Twin (VectorizedMECQueue) models multi-server M/M/c queuing theory dynamics for edge server computing clusters deployed on UAVs or Base Stations.

M/M/c Queuing Dynamics

Each Mobile Edge Computing (MEC) node operates \(c\) parallel CPU/GPU processing cores with individual service rate \(\mu = f_{\text{cpu}} / s_{\text{task}}\) (tasks/sec).

For a aggregate task arrival rate \(\lambda\), server traffic intensity \(\rho\) is defined as:

\[\rho = \frac{\lambda}{c \cdot \mu}\]

For queue stability, \(\rho < 1\).

Erlang-C Queue Waiting Probability

The probability \(P_q\) that an offloaded task must wait in the queue before processing follows the Erlang-C formula:

\[P_q = \frac{\frac{(c\rho)^c}{c!} \frac{1}{1-\rho}}{\sum_{k=0}^{c-1} \frac{(c\rho)^k}{k!} + \frac{(c\rho)^c}{c!} \frac{1}{1-\rho}}\]

Average Waiting & Execution Delay

The average queuing wait time \(W_q\) and total task response delay \(T_{\text{total}}\) are:

\[W_q = \frac{P_q}{c\mu - \lambda}\]

\[T_{\text{total}} = T_{\text{comm}} + W_q + \frac{1}{\mu}\]

where \(T_{\text{comm}} = D_{\text{task}} / R_{ij}\) is the radio transmission delay over the mmWave link.

Vectorized PyTorch Implementation

VectorizedMECQueue calculates queuing metrics across thousands of MEC servers simultaneously:

import torch
from urbanmarl.models.mec_queue import VectorizedMECQueue

queue_model = VectorizedMECQueue(
    num_servers=4,
    service_rate=100.0,  # tasks per second
    device="cuda"
)

# Compute queuing delays for batched arrival rates (B, N_mec)
delays, waiting_times, queue_lengths = queue_model.compute_delays(
    arrival_rates=lambda_matrix,
    data_rates=transmission_rates,
    task_sizes=task_sizes,
)