1.      Introduction

One Day International (ODI) cricket is a limited-overs format that was introduced as a faster-paced alternative to Test cricket. The first official ODI match took place on January 5, 1971, between Australia and England at the Melbourne Cricket Ground, as a result of a rain-interrupted Test match. The match consisted of 40 eight-ball overs per side, a format that later evolved to 50 six-ball overs per side, which remains the standard today. ODI cricket gained popularity quickly due to its shorter duration, allowing for a full match to be completed in a single day. The format introduced new

strategies, such as aggressive batting and tactical field placements, which added excitement to the game. The first ODI World Cup was held in 1975 in England, marking the start of a major international tournament that would eventually become one of the most-watched sporting events globally. Over the decades, ODIs have undergone several changes, including the introduction of colored clothing, day-night matches, and new fielding restrictions, making the format more appealing to a broader audience. Today, ODI cricket remains a cornerstone of international cricket, blending the endurance of Test cricket with the fast-paced action of Twenty20, making it a highly strategic and fan-favorite format[1].

queuing theory

In the context of One Day International (ODI) cricket, the arrival of batsmen and their service on the cricket pitch can be effectively modeled using queuing theory. Each pair of batsmen at the crease is seen as a "customer" being served by the cricket pitch, which acts as the "server." As soon as a wicket falls, the next batsman in line, or the "waiting customer," arrives to take the place of the outgoing batsman, ensuring a continuous flow of players during the innings[2,3]. The service time of each batsman can vary, depending on factors such as their skill level, the bowling attack, and game conditions. The utilization factor of the pitch, denoted by ρ (rho), represents the proportion of time the pitch is actively in use by the batting pair. When ρ = 1, it means that the pitch is fully utilized, with no idle time, indicating a continuous flow of batsmen and uninterrupted gameplay. In this scenario, the arrival rate of new batsmen matches the service rate, reflecting a balanced system where every incoming batsman immediately takes the place of the outgoing one.

2.       Literature Review

Queuing theory, originally developed to study systems involving waiting lines, has found diverse applications across fields, including sports analytics. Ghosal and Prakash (2025)[4], reviewed on International Cricket Council: rethinking growth strategy of cricket. In conclusion, the case equips students with a comprehensive understanding of strategic management within global sports industries, emphasizing growth, governance, and diversification. It further enables them to critically evaluate ICC’s strategic choices while formulating practical recommendations for sustained international success. Philipson (2024)[5]. A truncated mean-parameterized Conway-Maxwell-Poisson model for the analysis of Test match bowlers. Statistical Modelling. In conclusion, the truncated mean-parameterized Conway-Maxwell-Poisson model provides a robust and flexible framework for evaluating Test match bowlers, overcoming the limitations of traditional metrics. By incorporating temporal effects and Bayesian inference, it delivers more accurate rankings and deeper insights into player ability across cricket’s history. Halabi et al. (2023)[6], worked on the Financial Impact of World Series Cricket on Australia’s State Cricket Associations, 1974–1982. In conclusion, World Series Cricket initially caused severe financial and relational disruption for state associations, but its aftermath catalyzed structural reforms, stronger revenues, and an eventual strengthening of ties between the ACB and WSC. Ancker and Gafarian (1962) laid the foundation for queuing theory, providing the basic principles used in analyzing systems where customers wait for service[1]. These early models were generalized by Kendall (1953), who introduced stochastic processes to queuing systems, an approach that remains relevant when modeling unpredictable elements in sports, such as the fall of wickets in cricket[7]. In sports applications, Knott (1980) was one of the early contributors, highlighting the potential for queuing theory to model dynamic interactions within sports environments[8]. His work demonstrated that such models could be used to understand game flow and outcomes. Similarly, Skinner and Freeman (2009) applied queuing theory to penalty shootouts in soccer[9], illustrating its ability to predict high-pressure game situations. This approach is relevant to ODI cricket, where queuing models can predict key moments like when wickets will fall. Research by Katti and Khare (1992) extended these ideas, specifically discussing the role of queuing models in various sports[10]. They explored how the arrival of batsmen (customers) and the cricket pitch (server) can be modeled to predict the outcomes of different game scenarios. Jackson (1957) further developed queuing theory by introducing networks of waiting lines, which is useful in cricket when considering phases of the game, such as batting partnerships and the role of the next batsman waiting to play[11]. Specific to cricket, Anwar and Khan (2017) applied queuing models to analyze match outcomes and the impact of game interruptions, such as rain or wickets lost[3]. Their work provides insights into how the efficiency of gameplay, represented by the server utilization factor (ρ), influences whether a match ends in a tie or no result. This study aligns with the research by Sinha and Chatterjee (2010), who modeled cricket matches using queuing systems to predict game outcomes based on varying conditions[12]. Moreover, Bose and Basu (2008) focused on batting strategies in cricket using queuing theory, analyzing how batsmen pairings impact the progression of the game[13]. They applied this model to forecast when a batting team is more likely to build momentum or lose wickets, crucial for optimizing decision-making during ODIs. Akhtar and Scarf (2012) also explored forecasting in cricket, using dynamic linear models, which share underlying principles with queuing theory, particularly in managing stochastic, unpredictable elements like weather and match flow.

Broader applications of queuing theory, like the work by Pruyn and Smidts (1998) and Kozar and Holowczak (1984), though centered on service industries and sports scheduling, respectively, offer valuable insights for cricket[14-15]. These studies show that understanding the server (in this case, the cricket pitch) utilization and customer arrival patterns (batsmen coming to the crease) can provide essential predictions about game flow, delays, and outcomes in cricket.

2.1 Literature Gap

 The study by Verma and Gangeshwer (2022) applies the M/M/1 queuing model to IPL (T-20) but overlooks real-world complexities like multi-server systems, dynamic arrival rates, and player-specific factors[16]. Limited data validation, reliance on basic queuing theory, and a lack of integration with modern analytics highlight research gaps. Extending the model to other cricket formats, incorporating advanced tools, and analyzing decision-making impacts could enhance applicability and insights, addressing unexplored areas for future research in sports analytics.

2.2 Novelty of the work

In this paper we apply the M/M/1 queuing model to analyze One Day Internationals (ODIs), providing insights into batting partnerships, server utilization, and match outcomes. It will introduce probabilities for tied matches, no results, and winning outcomes validated against historical ODI data, revealing likelihood of ties or no results and emphasizing the competitive and unpredictable nature of ODIs.

3. Analysis of ODI Matches Using M/M/1 Queuing Systems

3.1 M/M/1 Model is a queuing model in which there is one server (the cricket pitch) and finite queue capacity. This model assumes

·         Arrival rate (λ): The rate at which new batsmen come to the crease, which happens after each wicket falls.

·         Service rate (μ): The rate at which a pair of batsmen stays at the crease before a wicket is taken.

This single-server model assumes a limited system capacity, where the maximum number of customers (or wickets) represents the total wickets that can fall in an ODI innings. Each innings has a finite queue (Let N=11), with arrivals corresponding to new batsmen coming to the crease, and service representing the time a pair of batsmen spends batting before a wicket is taken. The arrival rate and service rate quantify how frequently wickets fall and how long each pair stays at the crease, helping analyze match progression.

Probability of n cricketer’s in the system are

 The Probability of no cricketers in the system

We know that sum of probability is one so,

Hence the probability of n customer in the system is . Here the value of  never be less than one because arrivals have limit N.

The expected number of cricketers in the system is determined by

In ODI Context model, we let

Expected number of wickets in each innings is Near to 6 we can Assume.

The relationship between the average number of customers in a system ( ) and the average time a customer spends in the system ( ), as well as between the average number of customers in the queue ( ​) and the average time a customer spends waiting in the queue ( ​), is described by Little's formula. It is expressed as:   and .

Let ​ represents the effective arrival rate to the system. This rate is equal to the actual arrival rate  when all arriving customers are able to join the system. However, if some customers are unable to join because the system is full, then

In the context of the ODI model, if the system has limitations on accepting customers, the effective arrival rate  rather than the actual arrival rate , becomes the critical parameter. The effective arrival rate can be determined by analyzing the schematic diagram, where customers arrive at the source at a rate of  customers per hour.​

An arriving cricketers may enter the system or will be lost with rates  or  that is . A cricketer will be turned away from the system if the system already has its maximum capacity of cricketers.

     By def.

In other words, the average number of cricketers (wickets) in the queue is 5.09. This implies that, on average, approximately 5 batsmen are waiting in line during each innings. By definition, the difference between the average number of customers in the system and the average number of customers in the queue equals the average number of servers actively serving customers.

(near to 1). This means there is only one server actively serving customers.

4.       Estimation of Chances

Before the start of an ODI match, a coin is flipped to determine which team will bat first. We define the scenarios of batting in the first innings and batting in the second innings. Let  represent the event where the team bats in the first innings, and  represent the event where the team bats in the second innings. Each team has same probability. Additionally, we define the scenario where the first innings is completed, regardless of the number of wickets lost during the innings, along with the outcomes of the team winning in the first innings or second innings.

Let  represent the event where the first innings concludes, regardless of the number of wickets lost. Therefore . If fewer than wickets fall in the first innings, is still considered as, since it does not depend on the number of wickets lost. Let  represent the event where the team wins in the first innings, and  represent the event where the team wins in the second innings.

The likelihood of the team securing a win in the second innings is

The likelihood of the team achieving a win in the first innings is.

We now define the scenario where the ODI match ends in a tie or has no result (NR).

Let T represent the event where the ODI match results in a tie or has no result (NR). The likelihood of the match ending in a tie is

We note that the total probability is 1. This means the sum of the probabilities of a team winning in the first innings, a team winning in the second innings, and the match ending in a tie or having no result (NR) equals 1. Therefore, the probability of either team winning an ODI match is

We can also say, 4.54% of One Day Internationals (ODIs) end in a tie or no result (NR), while 95.45% conclude with a win for one of the teams. The chance of having no batsmen on the pitch in a single innings is

Table 1:- Actual ODI Record on January 1980 to December 2024.

 Exclude records of teams with a short ODI span (e.g., one, two, four, or a few years).

·                 Exclude teams with no tied matches (e.g., USA, UAE, Namibia, Hong Kong, Africa XI, Asia XI).

·         Only include teams with a significant and consistent duration in international cricket.

·         Include matches that were abandoned without a ball being bowled in the records.

This observation highlights that tied or no-result (NR) matches constitute a notable proportion of ODIs, indicating the impact of unpredictable factors such as weather interruptions or closely contested games. The percentage being higher than expected emphasizes the role of external elements and the competitive nature of international cricket.

Table 2 Overall ODI Match Statistics and Outcome Probabilities

At the beginning of an ODI match, there is no prior information about whether the match will end in a tie or have no result. Therefore, we focus on two possible outcomes: either the team batting first wins or the team batting second wins. The probabilities of these two outcomes are considered independent and equal. The expected value of a variable is represented by its mathematical expectation. Hence, the expected probability of the two events is determined accordingly.

We now validate this probability using the actual records of ODI matches, as outlined below.

Table-3 Consider the top 6 ODI teams in the word only.

From this table, we observe that 54.4% of ODIs have been won by the team batting first, considering the top ten ODI teams globally. The expected percentage is 47.72, indicating a significant difference compared to the actual winning percentage. This disparity confirms and validates our findings.

Table 4 Batting Order and Win Probabilities in ODIs

5.       Conclusion and Final Outcomes

The analysis of the M/M/1 queuing model in the context of ODIs shows that the expected number of wickets per innings is around six, with about five batsmen waiting in the queue on average, validating the single-server framework of the cricket pitch. The theoretical model predicted that approximately 95.45% of ODIs should produce a decisive result, while only 4.54% should end as tied or no result. However, actual historical data (1980–2024) reveals that around 5.9% of ODIs end without a result or as ties, slightly higher than the model prediction, largely due to external factors such as weather interruptions or evenly matched contests. Furthermore, while the expected probability of winning when batting first was 47.7%, the real-world data from the top ODI teams indicates a significantly higher rate of 54.4%, highlighting the practical advantage of batting first. This divergence underscores that while the queuing model captures the structural flow of wickets and outcomes, real cricket outcomes are shaped by broader contextual factors including playing conditions, strategies, and external uncertainties.

This study successfully demonstrates the application of queuing theory to analyse One Day Internationals (ODIs) in cricket, offering a novel statistical framework to assess match dynamics and outcomes. By treating the cricket pitch as a server and batsmen as customers, the M/M/1 queuing model effectively captures the flow of players and game progression during an ODI match.

The research highlights several key findings:

Utilization of the Cricket Pitch: The server (pitch) utilization factor, denoted by ρ, is an essential metric. A ρ value close to 1 reflects continuous gameplay, aligning with the observation of uninterrupted batsman-pitch interactions during ODIs.

Probabilities of Match Outcomes: The likelihood of a match ending in a tie or having no result (NR) was calculated to be 4.54%, while 95.45% of ODIs conclude with a clear winner. This reinforces the competitive nature of ODIs and the role of external factors such as weather in determining match outcomes.

Performance Insights: The expected number of wickets per innings, approximately 6, and the average number of batsmen in the queue, around 5, provide a quantitative understanding of team performance under standard match conditions.

Validation through Historical Data: By comparing theoretical probabilities with actual ODI records from January 1980 to December 2024, the study validates its model. The predicted win percentage for teams batting first closely aligns with real-world observations, demonstrating the model's robustness.

Implications for Strategy: The findings offer actionable insights for optimizing batting strategies and game management, aiding teams and analysts in decision-making processes. For example, understanding server utilization can inform line-up adjustments and pacing of runs.

This research provides a pioneering methodology for applying queuing theory in cricket analytics, emphasizing its potential to enhance both theoretical understanding and practical applications. The model serves as a valuable tool for cricket analysts, coaches, and teams to evaluate game dynamics, predict match outcomes, and refine strategies. Future work could extend this framework to incorporate additional variables, such as player fatigue or real-time match disruptions, for a more comprehensive analysis.