  We present a generalized hypercube queuing model, building upon the original
model by Larson 1974, focusing on its application to overlapping service
regions such as police beats. To design a service region, we need to capture
the workload and police car operation, a type of mobile server. The traditional
hypercube queuing model excels in capturing systems' dynamics with light
traffic, as it primarily considers whether each server is busy or idle.
However, contemporary service operations often experience saturation, in which
each server in the system can only process a subset of calls and a queue in
front of each server is allowed. Hence, the simple binary status for each
server becomes inadequate, prompting the need for a more intricate state space
analysis. Our proposed model addresses this problem using a Markov model with a
large state space represented by non-negative integer-valued vectors. By
leveraging the sparsity structure of the transition matrix, where transitions
occur between states whose vectors differ by one in the $\ell_1$ distance, we
can solve the steady-state distribution of states efficiently. This solution
can then be used to evaluate general performance metrics for the service
system. We validate our model's effectiveness through simulations of various
artificial service systems. We also apply our model to the Atlanta police
operation system, which faces challenges such as increased workload,
significant staff shortages, and the impact of boundary effects among crime
incidents. Using real 911 calls-for-service data, our analysis indicates that
the police operations system with permitted overlapping patrols can
significantly mitigate these problems, leading to more effective police force
deployment.
