Queueing networks for supporting container storage and retrieval

1. Introduction

In a container terminal of pure transshipment, an optimal management of stacking and retrieval operations at the storage yard has to be pursued as a major goal of system performance. It provides the key to guarantee a timely coordination of different container handling equipment and policies involved in the vessel discharge/loading (D/L) process. Clearly, an inefficient management of yard capacity and internal container transfer can often lead port operators to face serious and complex operational challenges (Carlos et al., 2014), especially when both import and export flows need to be handled concurrently.

The basic measure of performance for a well-performing D/L process relies on the timely and seamless flow of containers between the interacting terminal subsystems that operate at different time scales. The (average) quay crane productivity and/or the number of cranes allocated to an individual vessel is a target that is usually fixed by contractual conditions between the terminal operator and any given shipping line. The former party may have a degree of freedom in changing the number of assigned cranes and shuttle vehicles over the working shifts during which a specific vessel is berthed. This may be carried out by monitoring the status of the vessel's related D/L process in which activity progress is basically affected by both container transfer between the quay and yard areas and container stacking/retrieval operations on a specific part of the storage yard. For this reason, managing the path of shuttle vehicles through the transfer area and avoiding congestion during travel time as well as service blocking and resource starvation is mandatory.

Feeding, in a timely manner, both the quay and the container storage yard is crucial for achieving a time-effective configuration for the logistic process at hand. Queueing networks appear as the natural conceptual model at the basis of any modeling effort aimed to produce an effective time-oriented tool for performance evaluation. After some decades of research efforts (Dragović et al., 2017; Legato and Mazza, 2020), the adoption of discrete-event simulation (DES) rather than analytical queueing approximations has become a common choice for practice-oriented studies. Unfortunately, it often occurs that simulation users are attracted by the possibility of providing very detailed representations of their systems. So, they are faced with time-consuming activities for model development, verification, execution and statistical analysis of the output from a number of simulation runs which, let alone, is not easy to decide. This stated, our research effort addresses the challenge of proposing a two-level queueing-based hierarchical model. Simulation is used for the one or more subsystems at the inner level, while analytical solutions can be pursued for the outer model. Specifically, the outer model acts as a simplified queueing network where some artificial pure delay stations are inserted to receive the defining parameters from the results of the simulation carried out for the inner model. This allows the user to repeatedly work with the whole detailed fine-grained simulator once it has been developed, verified and validated. What-if experiments on the subsystem of interest can be performed by quickly screening and selecting candidate alternatives of subsystem (re)design. Hence, expensive and detailed simulation of the integrated conceptual queueing network will be reconsidered only to refine policies and resource allocation schema for a few (re)configurations.

Turning our attention to the model, generally speaking, container operations on the yard area of a human-operated transshipment hub are triggered by the activities taking place in the bordering quay and transfer subsystems. The occurrence of container discharge on the quay first calls for container transfer and then storage on the yard; vice versa, container loading on the quay first requires container retrieval from the yard and transfer to the quay afterward. As a result, interaction among these areas is provided by the (hopefully) seamless flow of containers generated by straddle carrier vehicles (SCs). While performing container handling and transfer operations, SCs are viewed as a finite population of customers performing round trips in a closed queueing network. This network features a central pure delay station acting as a finite source of customers for both the quay subsystem and the yard subsystem and two subsystems each of which consists in its own subset of single-server stations under a first-come-first-served (FCFS) service discipline. A sample network integrating the quay and the yard subsystems is shown in Figure 1 (Legato and Mazza, 2022).

Given the above network, we wish to contribute in widening the comprehension of its potentialities for the performance analysis of the real system of interest when having to account for strategic decision-making on (re)designing and managing the yard area organized in parallel storage rows accessed by human-operated SCs. To do so, we investigate the practical accuracy of combining analytical mean value analysis (MVA) with a detailed event-based simulation for computing system throughput and time duration of round trips, as well as queue lengths and waiting times arising at internal congestion points. This is our contribution to the practice of applying queueing networks for time-effective decision-making in container terminals.

The rest of the paper is organized as follows. The literature on the use of analytically solved queueing networks for the yard area is reviewed in Section 2. An MVA-based approximation for solving the queueing network that models container stacking/retrieval operations on the yard of a human-operated maritime terminal is proposed in Section 3. How simulation can support the above MVA approach is described in Section 4 and illustrated with numerical experiments. Conclusions are drawn in Section 5.

2. Literature review

Any issue about efficient yard management in container terminals would be incomplete if it did not mention the support provided by the use of operations research methods and models, for some time now (Steenken et al., 2004). In order to allow for more informed decision-making, timely problem-solving and improved efficiency in both operations planning and management, mathematical models and solution approaches have been adopted for describing, controlling and optimizing yard organization and practices. Among these, queueing-focused literature has often delivered analytical models that, if well tuned, can be pursued to produce an exact or, in most cases, a good approximation of the metrics of the terminal's (sub)system for a variety of sets of input parameters. To name a few, a semi-open queueing network model with bulk arrivals (by train), shared stack crane resources and multiclass containers is used in the study by Roy et al. (2022) to analyze the impact of prioritizing in stacking and internal transport handling the containers delivered by train. A closed-queueing network is proposed in the study by Xiang et al. (2022) to estimate the performance of an automated container terminal under traffic congestion, unbalanced task assignment, container batch arrival and different berth and yard layouts. An M/M/c queueing formulation is used in the study by Hasani Goodarzi et al. (2022) to model a cross-docking yard and focus attention on the receiving doors. A Jackson-type network approach is presented in the study by Ansorena (2020) to analyze port operations from vessel to yard. Focus is on both the berthing line and the service level at the storage yard. Batch arrivals of containers at a storage yard are considered in the study by Meštrović et al. (2018) with a multiserver queue M^X/M/c, under c yard cranes. Numerical experiments are returned for a different number of cranes which varies from 1 to 3.

On the other hand, complexity and many nonstandard operational features characterize the yard and the logistic processes carried out therein. As a result, an analytical-based solution approach of a yard-related queueing model is likely to be not accurate. This might lead to using the queueing model as a mere conceptual representation, while the analytical solution might be pursued for computational convenience alone.

For the reasons mentioned beforehand, the analysis of nontractable queueing-based models is susceptible to the use of more cost-intensive and/or time-consuming solution approaches. For example, a simulation-based truck appointment mechanism is proposed in the study by Shao et al. (2022), in conjunction with a DES model to balance truck arrival peak and reduce truck turnaround time via a tristage queueing system, while considering the interference of yard occupation brought by vessel discharging/loading operations. An integrated queueing network is presented in the study by Legato and Mazza (2018) as the natural modeling paradigm for a decision support system aimed at highlighting and quantifying queueing-related phenomena experienced in real practice by container transfer and handling equipment on both the quay and yard. The traffic system in an automated container terminal is modeled in the study by Zhou et al. (2017) as a network of servers that represents both paths and junctions. Rather than using an analytical queueing model, they resort to simulation to assess how traffic is affected by the nature of the job sequence in the specific container terminal, the number of vehicles deployed and the respective yard planning strategies. An algorithm is proposed in the study by Singgih et al. (2016) as the combination of a modified Dijkstra's algorithm for finding the shortest quay–yard time paths and a queueing theory for calculating the waiting times during the travel.

Unlike the previous efforts in the literature, our contribution is based on the idea of leveraging the combination of queueing and simulation as complimentary approaches to provide quality and time-effective solutions in logistic platforms.

3. Mean value analysis of queueing networks

Modeling daily practices in a logistic platform requires organizing and controlling internal handling and transfer operations in order to pursue both system-oriented and customer-oriented performance targets. Mean values for both system throughput and customer waiting time, with reference to completed operations, are the two typical indicators. To this purpose, an approximate, yet fast evaluation of the above performance measures is well appreciated. This may be pursued by the analytical solution of a queueing-based conceptual network model because it allows the analysis of internal points of congestion (service stations) throughout the logistic process of interest.

Focusing on a generic queueing network, nowadays practitioners could even resort to the so-called multiclass population of customers, where all the customers are partitioned into multiple groups. Each group is characterized by its own routing probabilities among the system resources, as well as its own service duration and service discipline at any given service station. On the other hand, a single-class network, such as the one focused by the queueing network in Figure 1, is the straightforward first-step model to provide for a system-oriented performance analysis.

By adopting a single-class or a multiclass closed network of the Baskett-Chandy-Muntz-Palacios (BCMP) type (Baskett et al., 1975), one could provide an exact solution to a model where performance measures may be computed exactly, but only once that several real features are relaxed. This is because the following set of simplifying assumptions is required in order for the exact solution to hold in the analytically tractable “product form” (Bolch et al., 2006):

1. Customers are routed among service stations according to a discrete-time Markov chain embedded at the instant of each customer departure from any given station;

2. Customers belonging to different classes are routed among service stations according to different Markov chains;

3. There are no restrictions on admitting customers to the queue upon their arrival, except for the case of adopting a rejection policy featuring random re-routing;

4. Service stations must feature either single servers with a fixed or load-dependent service rate and/or (identical) multiple servers, but with fixed service rates and, in no case, service interruptions nor server vacations are allowed;

5. Customers can be serviced in the FCFS order (besides random order and last-come-first-served), but the same exponential distribution is obliged for service times to customers belonging to different classes;

6. No priority disciplines for preemptive or non-preemptive services are allowed among customer classes.

Expected performance measures, such as throughputs and waiting times in queue, can be obtained for each service station starting by the MVA (Reiser and Lavenberg, 1980) algorithm. MVA is very easy to implement, in the version covering just a single-class population under a single fixed-rate server at any given station and one or more pure delay (infinite-servers) stations. Despite its popularity among the original community of computer scientists among which it was developed, MVA still needs renewed research efforts to deeper understand the extent to which the underlying “product-form solution” and the so called “arrival theorem” (Lavenberg and Reiser, 1980; Sevcick and Mitrani, 1981) are robust in practice and unavoidable for the theory underlying the exact solution of Markovian queueing networks. After 40 years, the extent to which a service station within the network can be treated as an external one and analyzed in isolation as subjected to a specific stochastic arrival process (state-dependent or not) is still a rather open research topic. The characterization of the stochastic flow of customers among the stations of a closed queueing network with an arbitrary topology and one or more return paths to the station just exited or previously exited by customers during the round trip between their (fixed) origin-destination stations is still an open problem. Despite this, scarcely justified decomposition-based approaches oriented to get an approximated solution of a non-product-form queueing network appear as the unique methods available for practitioners interested in fast, analytical solutions for practical performance evaluation studies in logistics.

4. Supporting MVA-based approximation by simulation

Performance evaluation studies on queueing stations and networks under several difficult features have been addressed in the studies by Dragović et al. (2006, 2012) and Legato and Mazza (2020) by approximations and simulation. Here we focus on the storage/retrieval operations occurring in the yard subsystem and embrace a deeper modeling prospective to capture some real rules of container handling and SC movements. To this purpose, the yard-related subnetwork of the queueing network model in Figure 1 is now analyzed by means of a closed network featuring two artificial pure delay stations, M peripheral service stations and a population of N SCs, as shown in Figure 2.

The SCs circulate alternating between the delay stations and the yard subnetwork. Here we do not pursue the classical flow-equivalent server representation of the berth and transfer areas subnetwork, under exponential service times, resulting from a Norton theorem–based reduction (Balsamo and Iazeolla, 1982) of any product-form subnetwork. In the current network representation portrayed by Figure 2, we adopt two artificial pure delay stations. In the first pure delay station (i.e. quay subsystem), in principle, each delay server should account for the duration of the activity under the quay crane which requires synchronization between the arrival of a vehicle and the availability of a container or buffer space for vessel discharge or loading, respectively. The second delay station (i.e. transfer subsystem) serves the purpose of modeling the number of vehicles simultaneously moving to/from the yard, while not disregarding the mutual dependencies arising among these vehicles (i.e. the active delay servers in the pure delay station). The average time duration for delay servers in both delay stations is provided by means of a model-driven simulator previously verified and validated (Legato and Mazza, 2018).

As for the yard subsystem and the operational rules applied therein, to perform handling and transfer operations, a vehicle must access a row in compliance with security measures that depend on both yard organization and technology. An SC may access a row if no other vehicle is already performing handling operations in that row or in either of the adjacent rows. As shown by the dotted arrows in Figure 2, by entering the central row to stack/retrieve a container, an SC issues a busy condition for that row and a so-called locking condition on the two adjacent rows. The locking condition yields dependency relations among the neighboring network stations, due to service prevention at a given station if an adjacent station has already started to deliver service. This stated, the resulting service time in the yard row may be represented by the stage-type diagram in Figure 3. Once again, simulation is used to provide the probability of locking and the average duration of the locking and container stacking/retrieval service (cit. op.).

As a result, this simulation-fed modeling approach first leads to restoring independency among service stations during service operations, a condition which is at the basis of the analytical tractability of any queueing network. It then paves the way for using analytical solutions, such as MVA-N, to carry out parameter-based analysis by systematically returning quantitative bound computation-oriented guidelines for (sub)system design, whether the (sub)system be existing or soon-to-be. As for the time complexity of the single-class MVA-based analytical solution for the overall network, observe that it is proportional to the product of M*N: it loops over the number of stations (inner level) and the number of circulating customers (outer level). So, when going from the four rows in Figure 2 to the actual number of rows in a real-life model, the computational burden remains acceptable.

5. Conclusions

A fast MVA of queueing networks is required for a first-order approximate performance analysis of complex logistic systems in order to overcome time-consuming simulation experiments in real environments. Usually, scheduling policies and allocation schema are evaluated and (re)organized to pursue a time-effective system management. After more than 40 years of modeling efforts, one has to recognize that congestion arising from resource sharing under the occurrence of resource locking, starvation and blocking phenomena cannot be captured by heuristic adjustments on analytical formulas derived for product-form queueing networks. So, a revisitation of some analytical results for product-form queueing networks and related approximations covering nonexponential service times have been investigated. Then, a hierarchical combination of simulation with analytical solution methods may offer satisfactory achievements. Partial results from DES have been fed into a simplified analytical model for storage and retrieval operations of containers in a maritime transshipment hub. Numerical results encourage more research efforts along this avenue to the aim of specializing the classical simulation metamodeling approach through the adoption of fast-to-solve, yet accurate analytical queueing networks.