1.1 Research Background

Improving healthcare systems has become a strategic priority worldwide. These systems, which encompass a broad range of human and material resources, involve complex interactions among different care groups [¹-³]. They are characterized by a high degree of variability and uncertainty, making it challenging to represent and understand their functioning [⁴-⁸]. With the continuous increase in the global population and the emergence of new diseases with varying symptoms, as illustrated by the COVID-19 pandemic [⁹], the demand for healthcare services has significantly risen. This dynamic complicates the tasks of decision-makers and makes the complexity of healthcare systems increasingly evident [¹⁰, ¹³].

Healthcare systems face several complex issues. Patient waiting time management is one of the most pressing problems. Prolonged delays can not only decrease patient satisfaction but also have serious consequences on their health [¹¹]. At the same time, optimizing resource utilization rates is crucial to avoid overloads and inefficiencies that can lead to additional costs and a decline in care quality [¹]. Managing patient flows, especially during peak periods, is another major challenge. Poor flow management can result in congestion, extended waiting times, and a decrease in the quality of care [¹²]. These challenges are exacerbated by the increasing complexity of healthcare systems, requiring effective and rigorous management strategies to ensure their proper functioning and reliability [¹³].

Modeling and simulation prove to be valuable tools for overcoming these challenges and solving these problems [¹⁴]. They help forecast future resource needs, analyze and test different scenarios, identify critical points and bottlenecks, and develop strategies to manage unexpected situations without risking negative impacts in the real world [¹⁵-¹⁶]. Modeling involves creating a conceptual model of a problem within a system, based on theories or observations [¹⁷-¹⁹]. This conceptual model must include various essential elements, such as inputs, outputs, content, boundaries, and objectives [²⁰-²¹]. Once developed, this model is translated into computer software to be simulated under different conditions, allowing healthcare professionals to analyze and better understand the system's behavior in a real-world context [¹, ²²-²⁴].

1.2 Motivation and Problem Definition

In the field of healthcare, various modeling methods are used to analyze and improve healthcare systems, including:

Although these methods are commonly used to address healthcare issues and provide powerful tools for analyzing and improving healthcare systems, they have limitations in terms of formal precision [²⁵-³²] and in verifying and managing errors related to the specification of their requirements [²⁵-⁴⁰]. In this context, using Hoare Logic provides a formal approach to specify and formally verify the requirements of healthcare systems. By integrating formal inference rules, the proposed approach aims to enhance the rigor and reliability of simulations, thereby better managing the complexity of systems and improving their performance.

Hoare Logic [⁴¹] is a formal method used for specifying and verifying program properties in a rigorous and mathematical manner. In the context of healthcare, the contribution of Hoare Logic is to provide a formal foundation to ensure that systems adhere to certain essential properties and critical invariants. By integrating Hoare Logic inference rules into the simulation model, this approach is useful for identifying inefficiencies, improving the accuracy of simulation results, ensuring the reliability of operations, and facilitating the decision-making process.

The main contribution of this research paper is to prove how Hoare Logic can be applied to improve healthcare systems performance and support healthcare professionals' decision-making.

1.3 Contribution

The main contributions of our research paper are as follows:

First, we propose a new Hoare Logic based approach by using inference rules specifically adapted for the formal specification and verification of healthcare systems requirements, with a particular focus on the dental department of an Algerian hospital. Our research work is original and makes a significant contribution in health domain by proving the practical application of Hoare Logic and inference rules for the specification, verification, and performance improvement of healthcare systems. Through our proposed case study, we illustrate how the new approach can be used to optimize resource utilization, identify inefficiencies, and minimize patient waiting times while ensuring the reliability and safety of healthcare systems.

Secondly, we propose a simulation Hoare Logic inference rules based-tool to enhance the performance of healthcare systems and support healthcare professionals' decision-making.

The remainder of the paper is structured as follows: Section 2 reviews related work on modeling healthcare systems. Section 3 classifies various healthcare systems modeling methods according to their advantages and limitations. Section 4 details the proposed Hoare Logic based-approach, while Section 5 presents the inference rules used to specify and verify healthcare systems requirements. Section 6 is dedicated to the simulation of a case study, with a discussion of the simulation results in Section 7, and a proposal of solutions to improve performance in Section 8. Finally, Section 9 concludes the paper and suggests directions for future research.

2.1 Workflow Approach

In [²⁵], the authors developed a workflow model to analyze patient pathways in the pediatric department using real data. This model helps identify bottlenecks and overcrowding situations that contribute to increased service load. They were able to propose improvements to reduce delays and optimize care processes.

In [²⁶], authors built a workflow model to understand and improve the management of patients with uncomplicated sepsis, severe sepsis, and septic shock in an emergency department. Their model contributed to earlier detection and better management of sepsis in the emergency department, thereby speeding up care delivery.

In [²⁷], the authors proposed a workflow model to model the dental center system using data collected in the hospital environment. The accuracy of the model was tested in the real system, contributing to reduced patient waiting times and increased staff productivity.

In [²⁸], the authors proposed an approach to classify and analyze workflow interruptions in the emergency department. The results helped determine the correlations between workflow interruptions and professional stress, enhancing the understanding of the benefits and potential risks of interruptions in emergency work environments.

In [²⁹], the authors constructed a workflow model to design and describe patient flow in the hospital emergency department. Their model contributed to reduced patient waiting times and improved patient flow quality.

2.2 UML-Based Approach

In [³⁰], the authors used UML diagrams to capture key aspects of pediatric care pathways, facilitating comparison across different European Union countries within the Models of Child Health Appraised project. Use case diagrams allowed visualization of activities and actors involved, providing an overview of care processes.

In [³¹], the authors employed UML models to create various diagrams such as use case diagrams, class diagrams, sequence diagrams, and collaboration diagrams. These diagrams were used to address requirements for regular patient visits, hospitalizations, medication management, and other related tasks. This approach improved visualization and understanding of the complex interactions between different components of the healthcare system, thereby enhancing service efficiency and coordination.

In [³²], researchers utilized a UML activity diagram to model the surgical process while developing a probabilistic queuing network model to assess different configurations. This approach enabled the determination of system parameters and conducted simulations to improve surgical processes.

2.3 Agent-Based Modeling Approach

In [³³], the authors proposed a medical multi-agent system to enhance coordination and management within healthcare systems. This system uses intelligent agents to manage collaboration between departments, coordination among medical staff, scheduling of medical diagnoses, and patient information collection. By integrating these capabilities, the system developed by the authors aims to improve the quality of patient care by optimizing organizational processes and facilitating more effective management of resources and information.

In [³⁴], the authors proposed an agent-based simulation tool to evaluate rapid treatments in emergency services. The study compares static processes, which use fixed durations for daily operations, and dynamic processes, which adapt treatment based on current conditions in the service. The results provide valuable insights for implementing rapid treatment to reduce patient waiting times in emergency departments.

In [³⁵], the authors developed an agent-based method to model emergency services even with incomplete data. Their approach demonstrates the flexibility and effectiveness of multi-agent systems in managing complexities and uncertainties within emergency services.

In [³⁶], the authors developed an agent-based modeling framework to simulate the behavior of patients leaving an emergency department of a public hospital without being seen by a doctor. The study used cellular automata and agent-based modeling to evaluate the impact of four preventive policies. After applying these policies, an average reduction of 42.14% in the number of patients leaving without being seen and a 6.05% reduction in patient length of stay were observed, with the most effective policy being the rapid treatment of less critical patients. This approach provides a flexible tool for decision-makers to assess the relative impact of control strategies in emergency departments.

2.4 Petri Net Modeling Approach

In [³⁷], the author focuses on analyzing and improving the performance of emergency departments using a model based on hierarchical timed colored Petri nets. The study aims to enhance key performance indicators such as patient waiting times, length of stay, and resource utilization rates. By using historical data to validate the model, several improvement scenarios were tested, demonstrating potential reductions in waiting times and lengths of stay. The most promising scenarios were identified based on their effectiveness and cost.

In [³⁸], the authors use a colored Petri net to model an emergency department. The main objective of this study is to reduce patient waiting times and, consequently, their length of stay in the service. Through simulations of various improvement scenarios, they identify effective strategies to optimize service performance.

In [³⁹], the authors explore the improvement of patient treatment processes in cardiac clinics during the COVID-19 pandemic using timed colored Petri nets (TCPNs). The study aims to optimize patient waiting times and prepare hospitals to better manage the challenges posed by the pandemic. The authors simulated current clinic processes, identified bottlenecks, and proposed optimization scenarios. Their approach demonstrates how TCPNs can be applied to enhance workflows during a crisis.

In [⁴⁰], the authors address the management of emergency departments (EDs) with a focus on reducing patient waiting and stay times. The study proposes an approach combining modeling and optimization to improve resource utilization. The authors used a colored Petri net to model the system and genetic algorithms to determine the optimal number of resources needed. Simulations provided several models, each contributing to reducing both the length of stay and the time from door to physician.

4.1 Generic Model

The proposed model is based on three generic elements: the patient entity, the resource entity, and the treatment entity, which are interconnected through interactions. Each element is described by various attributes, and specific functions are also provided to manage each entity.

A hierarchical organization is adopted to allow communication between entities of the same level. Thus, the elements—patient, resource, and treatment—exchange information about the patient's treatment on the resources: each patient Pi undergoes one or more operations Tj; each resource Rk handles one or more operations Tj, and each operation Tj is handled by a single resource.

4.2 Simulation and Verification Process for Healthcare Systems

The proposed approach integrates Hoare Logic inference rules into the simulation model to specify and verify the properties of healthcare systems. The simulation and verification process is structured around the following steps:

First, the resources are initialized and prepared for use. Then, patients are managed according to the specified sequence of treatments upon their arrival. At time t, patient Pi arrives and checks the availability of resource Rj for the first operation T1. If resource Rj is occupied, patient Pi waits in the queue; otherwise, they begin the treatment. The patient then proceeds to operation T2, and so on until all operations are completed.

System properties such as the waiting time for each patient, overall simulation time, resource utilization rates, and resource inefficiency times are calculated throughout the simulation. The proposed inference rules are applied to detect potential errors, such as excessive waiting times, high utilization rates, and resource inactivity overruns, and to suggest adjustments aimed at improving system efficiency.

This process has been implemented through the development of a simulation tool that integrates the generic model, the proposed inference rules, and the simulation principles to enhance healthcare system performance and support healthcare professionals in their decision-making.

5.1 Hoare Logic

Hoare logic, sometimes called Floyd-Hoare logic, is a formal method defined by Tony Hoare in [⁴¹], is a formal method used for the specification, verification, and proof of computer programs. Its key concepts include Hoare triples, invariant predicates, and inference rules. Although Hoare logic was primarily developed for programming, it can be adapted and extended to be relevant in the specification and verification of complex systems.

A Hoare triple, of the form {P} C {Q}, describes a command C in relation to a precondition {P} (the state of the system before executing C) and a post-condition {Q} (the state of the system after executing C). These Hoare triples are relevant for the specification and verification of complex system requirements because they allow for the formal specification of conditions before and after the execution of an action. This can be used to describe system states, state transitions, and desired properties.

Invariant predicates, denoted {I}, represent properties that must remain constant throughout the execution of a simulation program. They are essential for modeling complex systems as they help ensure that certain important properties are maintained throughout execution, which is crucial for the stability and safety of complex systems.

Inference rules enable the proof of Hoare triples' validity by deriving them from other triples, thereby ensuring a rigorous verification of specifications.

Table 2 presents the list of elements of the healthcare system model, their attributes, and the abbreviations used in our work.

In the following three subsections, we will present the adapted inference rules designed to optimize resource utilization and minimize patient waiting times to ensure the reliability and safety of healthcare systems. Each assertion of these rules is crafted to verify a specific property before and after significant operations, such as patient intake or diagnostics. The pre-conditions and post-conditions defined in these assertions ensure that specified requirements are met, including patient waiting times, resource utilization rates, and inactivity periods, throughout the simulation. If a violation of an assertion occurs, a precise error message is generated to indicate the property that has not been satisfied.

5.2 Inference Rules for Verifying Patient Waiting Time

The waiting times for each patient are calculated to ensure adherence with the maximum constraint. Controlling patient waiting times is essential for improving overall system efficiency and ensuring a continuous flow without bottlenecks. The verification of patient waiting time is based on an inference rule composed of three assertion triplets: Assert 1, Assert 2, and Assert 3. The post-condition of the first triplet (Assert 1) serves as the pre-condition for the second triplet (Assert 2), while the post-condition of the second triplet (Assert 2) becomes the pre-condition for the third triplet (Assert 3). This rule formally specifies and verifies the treatment steps of a patient Pi, with a particular focus on respecting the maximum waiting time constraint, as detailed below.

Assert1: This assertion specifies the first step where Patient Pi arrives and finds Resource Rw occupied. At time V1, Patient Pi arrives with an initial waiting time V2 to receive Treatment Tk on Resource Rw. However, Patient Pi discovers that Resource Rw is currently occupied by Patient Pj. Consequently, from time V4 onward, Patient Pi begins to wait in the queue for Resource Rw:

Assert 2: This assertion specifies the second step where Patient Pi is waiting for Resource Rw to be released. At time V5, Patient Pj completes Treatment Tk on Resource Rw and releases the resource.

Assert 3: This assertion specifies the third step where Patient Pi begins treatment after waiting. At time V6, Patient Pi starts Treatment Tk on Resource Rw, following a cumulative waiting time of V7.

Applicability Conditions: The conditions (1,2,3) detail the necessary calculations within the three assertions (Assert 1, Assert 2, Assert 3) to verify that the maximum waiting time for Patient Pi is respected.

The new waiting time V7 for Patient Pi is calculated as follows: V7 = V2 - (V5 − V4) + the maximum preparation time of Resource Rw, where V2 represents the previous waiting time for Patient Pi, V5 denotes the completion time of Patient Pj’s treatment, and V4 marks the time when Patient Pi began waiting in the queue for Resource Rw. The cumulative waiting time V7 for Patient Pi must not exceed the maximum permissible waiting time. If the assertions are violated, a specific error message will be generated to indicate that the waiting time for Patient Pi has not been respected.

5.3 Inference Rules for Verifying Resource Utilization Rate

The utilization rate of each resource is measured to identify underutilized or overloaded resources. The verification of resource utilization relies on an inference rule composed of two triple assertions: Assert 4 and Assert 5. The post-condition of the first triplet (Assert 4) serves as the pre-condition for the second triplet (Assert 5). This rule formally specifies and verifies the treatment step of a patient Pi, with a particular focus on ensuring compliance with the utilization rate of Resource Rw, as follows:

Assert 4: This assertion specifies the first step where patient Pi begins their treatment after his arrival or waiting. At time V3, patient Pi starts treatment Tk on resource Rw, with an initial treatment time of V2.

Assert 5: This assertion specifies the second step where patient Pi completes their treatment on resource Rw. At time V5, patient Pi finishes treatment Tk on resource Rw, and the new cumulative treatment time for resource Rw becomes V4:

Applicability Conditions: The conditions (4,5, 6,7,8) detail the necessary calculations in the two assertions (Assert 4 and Assert 5) to verify that the maximum utilization rate for resource Rw is maintained.

5.4 Inference Rules for Verifying Resource Idle Times

The periods during which resources are not in use are measured to assess and reduce inefficiencies. Monitoring the periods when resources are unused or waiting is based on an inference rule composed of two triples: Assert 6 and Assert 7.

The post-condition of the first triplet (Assert 6) serves as the pre-condition for the second triplet (Assert 7).

This rule formally specifies and verifies the treatment scenario of a patient Pi, with a particular focus on ensuring the idle time of resource Rw is respected as follows.

Assert 6: This assertion specifies the first step where the first patient Pi completes their treatment Tk on the resource Rw at time V2:

Assert 7: This assertion specifies the second step where the next patient Pj begins their treatment after arrival or waiting. At time V4, the next patient Pj (after patient Pi) starts their treatment Tk on the resource Rw, with their cumulative inactive time being V5:

Applicability Conditions: The conditions (9), (10), and (11) detail the necessary calculations in the two assertions (Assert 6 and Assert 7) to verify that that the inactivity time for the resource Rw is respected:

6.1 Presentation of the Dental Department of an Algerian hospital

The healthcare system under consideration is the dental department located in a hospital situated in western Algeria. This department holds a significant position within the hospital. Dental medicine is a medical-surgical specialty that diagnoses and treats conditions of the oral cavity and the maxillofacial regions.

6.2 Data Model of the Dental Department

In the context of the generic model, the Patient entity includes 30 patients labeled P1 to P30, and the Resource entity includes 8 resources R1 to R8 corresponding to 7 doctors and 1 secretary. The Treatment entity includes all 8 treatments T1 to T8, representing the 5 specific operations of the dental department along with the secretarial function. These operations are Pathology (performed by 2 different doctors), Periodontology, Dentofacial Orthopaedics (ODF), Oral surgery, and Dental Prosthetics (performed by 2 different doctors).

The complete information about the department's resources (Fig. 2) includes the identifier (Id_Ri), name, type (human), resource status (free), and minimum and maximum preparation times. This time is set to 1 time unit (TU) for all resources. The detailed characteristics of the resources are modeled to simulate the healthcare system realistically. Resource constraints include a resource treatment rate not exceeding 95% and a maximum resource inefficiency time of 35 time units.

Fig. 2 Information on the Resources of the Dental Department 

Similarly, the eight treatments (Fig. 3) exchange information with the resources, including the treatment identification (Id_T) and the identification of the concerned resource (Id_R), the name of the treatment, as well as the minimum and maximum processing times.

Fig. 3 Information on the Treatments of the Dental Department 

According to the principle of communication between entities, the 30 patients (Fig. 4) exchange treatment information, such as the patient's arrival time, the number of operations performed for the patient's treatment, and the corresponding treatment sequence. For example, patient P1 undergoes treatments T8 and T1, according to the relevant operations. The patient constraint includes a maximum waiting time of 100 time units.

Fig. 4 Information on the Patients of the Department 

All patients are initially processed through operation T8, as they first interact with the secretary (resource R8), who handles their files. They are then attended to by a doctor (resources R1 to R7) who will perform a treatment Ti (i=1 to 8, with i ≠ 8) based on the type of care required. For example, patient P1 undergoes a pathology treatment (T8-T1), while patient P3 undergoes a periodontology treatment (T8-T2).

7.1 Performance Relative to Patients

7.1.1. Results

By applying the simulation principle, the evaluation of performance criteria for the 30 patients (Fig. 5) produced the following results concerning the waiting time (in time units) of the patients.

Fig. 5 Waiting Times of Patients in the Dental Department 

7.2 Performance Relative to Resources

7.2.1. Results

By applying the same simulation principle, the evaluation of the performance of the 8 resources (doctors and secretary) produced the following results (Fig. 6) concerning the utilization rate. The resource utilization rate was calculated at time 300 UT, corresponding to the end of the morning work period.

Fig. 6 Resource Utilization Rates in the Dental Department 

7.3 Reported Errors and Suggestions

8.1 Improvement in Work Time Performance

The simulation ended at 235 time units, reflecting a reduction of 132 time units compared to the previous scenario.

8.2 Improvement in Patient Performance

8.2.1. Results

The performance improvements for the 30 patients resulting from the application of our approach are illustrated by the results shown in Fig. 7. These results display the waiting times (in time units) for patients before and after implementing the inference rule detailed in subsection 5.2.

Fig. 7 Improved Waiting Times for Patients in the Dental Department 

8.3 Improvement in Resource Performance

8.3.1. Results

The enhanced performance resulting from the application of our approach for the 8 resources (doctors and secretary) is illustrated by the following results (see Fig. 8). These results show the utilization rates before and after applying the inference rules detailed in subsections 5.3 and 5.4. The resource utilization rates are calculated at time 300 time units, corresponding to the end of the morning work period.

Fig. 8 Improved Resource Utilization Rates in the Dental Department (calculated at time 300 UT)