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.

Judicial backlog; ruin theory; Cramér–Lundberg model; heavy-tailed distributions; Hill estimator; bidimensional risk model

1. Kenya Judiciary. State of the Judiciary and Administration of Justice Annual Report FY 2024/25. Nairobi: Judiciary of Kenya; 2024. [Google Scholar] [Crossref]

2. Mukherjee S. Heavy-tailed distributions in judicial case-disposition times. J Empir Leg Stud 2018;15(2):234–257. [Google Scholar] [Crossref]

3. Barabási A-L. The origin of bursts and heavy tails in human dynamics. Nature 2005;435:207–211. [Google Scholar] [Crossref]

4. Thorburn H. A simulation model of the NSW District Criminal Court with illustrative applications. Crime Justice Bull 2017;(204). NSW Bureau of Crime Statistics and Research. [Google Scholar] [Crossref]

5. Clarke DR, Metzler A. Estimating the delay of criminal trials: evidence from Canada. J Law Empir Anal 2025;2(1):33–59. [Google Scholar] [Crossref]

6. Doe F. Processing times in refugee tribunals: evidence of variability and delays. J Refug Stud 2022. [Google Scholar] [Crossref]

7. Adebayo T. Empirical analysis of case duration in Nigerian courts. Niger J Leg Stud 2020;12(2):88–104. [Google Scholar] [Crossref]

8. Hlophe M. Criminal case delays in South African courts: a stochastic perspective. S Afr J Crim Justice 2019;32(3):120–136. [Google Scholar] [Crossref]

9. Sibanda P. Backlogs and systemic inefficiencies in South African criminal courts. Afr Law Rev 2021;15(2):75–91. [Google Scholar] [Crossref]

10. Kandie J, Atitwa P, Nyaluke E. Perspectives on judicial efficiency: case study of predictive modeling in Kenyan high courts. Int J Court Adm 2025. [Google Scholar] [Crossref]

11. Makau J. Factors influencing management of case backlog in the judiciary in Kenya [MA thesis]. University of Nairobi; 2014. [Google Scholar] [Crossref]

12. Mulwa P. The effects of magistrates' transfer on the administration of justice in Kenya — a case study of courts in Nairobi City County [thesis]; 2020. [Google Scholar] [Crossref]

13. Embrechts P, Klüppelberg C, Mikosch T. Modelling Extremal Events for Insurance and Finance. Berlin: Springer; 1997. [Google Scholar] [Crossref]

14. Waters HR, Papatriandafylou A. Ruin probabilities allowing for delay in claims settlement. Insur Math Econ 1985;4(2):113–122. [Google Scholar] [Crossref]

15. Asmussen S, Albrecher H. Ruin Probabilities. 2nd ed. Singapore: World Scientific; 2010. [Google Scholar] [Crossref]

16. Ganesh A, Torrisi GL. A class of risk processes with delayed claims: ruin probability estimates under heavy-tail conditions. J Appl Probab 2016;53(3):750–759. [Google Scholar] [Crossref]

17. Da L, She K. A large deviation principle for stochastic hydrodynamical-type systems driven by multiplicative Lévy noise. Stoch Process Appl 2022;132:1–25. [Google Scholar] [Crossref]

18. Lu D, Yuan M. Asymptotic finite-time ruin probabilities for a bidimensional delay-claim risk model with subexponential claims. Methodol Comput Appl Probab 2022;24(4):2265–2286. [Google Scholar] [Crossref]

19. Li S. Random multiplicities in unidimensional surplus models. Insur Math Econ 2013;53(2):321–335. [Google Scholar] [Crossref]

20. Fu MC, Kou S. Heavy-tailed queues and risk modeling. Oper Res 2015;63(5):1117–1132. [Google Scholar] [Crossref]

21. Lundberg F. Approximerad Framställning av Sannolikhetsfunktionen. Uppsala: Almqvist & Wiksells; 1903. [Google Scholar] [Crossref]

22. Cramér H. On the Mathematical Risk Theory. Skandia Jubilee Volume; 1930. [Google Scholar] [Crossref]

23. Asmussen S. Applied Probability and Queues. 2nd ed. New York: Springer; 2003. [Google Scholar] [Crossref]

24. Niemeijer N, Spruit M. Stochastic modeling of court delays. Eur J Oper Res 2012;219(1):34–46. [Google Scholar] [Crossref]

25. Vermeulen P, et al. Delay, inefficiency, and perceived fairness in judicial systems. Law Soc Rev 2014;48(3):657–686. [Google Scholar] [Crossref]

26. Okoth F, Muli J. Impact of repeated adjournments on statutory timelines in Kenyan courts. Kenya Law J 2018;12(2):101–123. [Google Scholar] [Crossref]

27. Gichuhi P. Seasonality and long-memory in Kenyan judicial caseloads. Afr J Law Technol 2019;7(1):45–66. [Google Scholar] [Crossref]

28. Gindicha JY. Quantifying case flow efficiency in Kenya's Employment and Labour Relations Courts: a multi-stage queueing model approach. Int J Math Stat Invent 2025;13(3):01–09. [Google Scholar] [Crossref]

29. Marang'a MW. Does court type, size and employee satisfaction affect court speed? Hierarchical linear modelling with evidence from Kenya. J Law Policy Glob 2021;110:6–20. [Google Scholar] [Crossref]

30. Marang'a M, Kimalu P, Ochieng M. Effect of COVID-19 pandemic on resolution of cases in courts: the Kenyan Judiciary. Research Paper 2021;1. [Google Scholar] [Crossref]

31. Marang'a MW, Ouma D, Kosimbei G. Congestion in the justice system and private investments. J Poverty Invest Dev 2018;1(1):1–19. [Google Scholar] [Crossref]

32. Xu C, Wang K, Wu X. The finite-time ruin probability of a risk model with stochastic return and subexponential claim sizes. Commun Stat Theory Methods 2024;53(6):2194–2204. [Google Scholar] [Crossref]

33. Yang H, Geng Y, Wang Y. Multivariate surplus models with random delayed claims. Scand Actuar J 2014;2014(3):245–267. [Google Scholar] [Crossref]

• Assessment of the Role of Artificial Intelligence in Repositioning TVET for Economic Development in Nigeria
• Teachers’ Use of Assure Model Instructional Design on Learners’ Problem Solving Efficacy in Secondary Schools in Bungoma County, Kenya
• “E-Booksan Ang Kaalaman”: Development, Validation, and Utilization of Electronic Book in Academic Performance of Grade 9 Students in Social Studies
• Analyzing EFL University Students’ Academic Speaking Skills Through Self-Recorded Video Presentation
• Major Findings of The Study on Total Quality Management in Teachers’ Education Institutions (TEIs) In Assam – An Evaluative Study