Asymptotics for Backlog Probabilities in a Bidimensional Caseload Model with Randomly Delayed Cases: Evidence from Kenya's Magistrates' Courts

Persistent case backlogs in Kenya’s Magistrates’ Courts undermine timely access to justice, with these courts handling over 79% of pending cases nationwide. Existing measures such as average processing times and clearance rates offer limited insight into the extreme delays driving systemic congestion. This study developed a bidimensional stochastic surplus model jointly incorporating criminal and civil case streams with randomly delayed proceedings, extending the Cramér–Lundberg framework to integrate Poisson arrivals, heavy-tailed workload distributions, and stochastic adjournment delays. Maximum likelihood estimation identified appropriate distributions for workload and adjournment-gap duration, while the Hill estimator characterised tail behaviour; joint and marginal backlog probabilities were derived using asymptotic and large-deviations theory. Both streams were operationally unstable (θ_Criminal=-21.57; θ_Civil=-35.48), so stream-level backlog probability was reported as not applicable; stability would require capacity near 120 and 105 hearings per day, or arrival-rate cuts of about 25% and 50%, respectively. System-wide, adjournment durations showed heavier-than-exponential tails (α ̂_Civil=2.08, α ̂_Criminal=4.48), validating the polynomial decay result as an operative estimate for the civil stream and a stress-testing benchmark for the criminal stream. Scenario simulations identified the adjournment tail index as the dominant risk driver: shifting it from 3.5 to 1.8 raised marginal backlog probability by 1,678%, far exceeding doubling capacity (-9.3%) or arrival intensity (+13.9%). Effective mitigation therefore requires prioritising adjournment control over capacity expansion, offering evidence-based tools for congestion-threshold identification and backlog-reduction strategy in Kenya’s Judiciary.