Bifurcaciones locales

distribution. In this section, the implementation of the Coxian k-phased approxi-mation for a general distribution will follow the work from Altiok (1996) and Curry and Feldman (2011), by using the moment-matching approximations strategy. In

1 p 2 µ₂ (1-p)

µ₁

Fig. 6.1.: MGE 2-phased Transition Diagram with ¹₂ ≤ C² < ∞

their research, the LST (Laplace-Stieltjes transform) “of any distribution function can be approximated arbitrarily close by a rational function.” The main problem, they explain, is the lack of a method to determine the parameters and structure of the phase-type distribution.

Consider the transition diagram from Figure 6.1, the parameters needed for a MGE-2 include µ_i (mean service time per phase i) and p (probability of moving to the next phases), where the idea is to get arbitrarily close to the approximating distribution. In Section 7.2, the squared coefficient of variation, C², is discussed and the parameters are calculated using the the empirical data in order to be implemented in the simulation model. In the analytical model, the data is fitted to a 2-phase approximation to better handle the number of phases.

6.2.1 Research for Approximating Service Distributions

Authors like Ojah et al. (2002) and Bradbury (2010) coincide with Haralambides and Londono-Kent (2001) in their conclusion that “study of what actually happens at the border reveals significant time and cost inefficiencies in the border crossing process.” And the difficulty in this matter is discussed in Section 3.2. However, the research done by Ashur et al. (2001) use the Erlang distribution because it “is frequently used in queueing systems to represent service-time distributions in discrete systems simulation.”

Usually, phased-type distributions are used in models of stochastic characteristics.

Altiok (1996) discusses the use of MGE, often called Coxian distributions, which have been used in these type of analyses. Additionally, these distributions have been used in the analysis of manufacturing, computer and communication systems. When using these distributions, they are characterized by phases, that is, spending an exponentially distributed amount of time in each phase, and the key is determining the number of phases needed and the phase. Curry and Feldman (2011) and Altiok (1996) elaborate on the approximation of service times using MGE, with more details to follow the discussion in Section 6.4.

Whitt (2007) also used approximation methods to help set the staffing require-ments in service systems he was researching. His queueing model was a Mt/GI /st+ GI. His work is similar to this research of POEs, in that the “model is difficult to analyze mathematically, so that the staffing problem is challenging. However, there is one special case that is amazingly tractable: the Markovian M_t /M /s_t+ M model in which θ = µ.”

There are many papers dealing with the analysis of non-stationary queueing sys-tems, e.g., Choudhury et al. (1997), Ong and Taaffe (1988), Margolius (2005), and Margolius (2007), and they almost always begin with the Chapman-Kolmogorov for-ward equations which will be used for the analytical model. However, there are not many decision control problems using these formulations. Most of the queue control literature, e.g., Adusumilli and Hasenbein (2010), Ata (2006), and references in the research work of Ata (2006), deal with steady-state results for stationary queueing systems.

6.2.2 Analytical Model Benefits

Mathematical analysis is the basis of many research studies. The aim is to char-acterize the system’s behavior so that improvements can be validated, and in some cases, proven to work. Having the entire model described by a mathematical

struc-ture, for example the probability distribution for the number of jobs in the system (p_n = Pr{N = n}) is very desirable, particularly in modeling queueing systems.

Once that information is mathematically developed, the entire system can be char-acterized, and all the information about the its behavior can be computed.

When a model can be described mathematically, and an analytical solution to a mathematical model is available without being computationally inefficient, “it is usually desirable to study the model in this way, rather than via a simulation” (Law and Kelton, 1991). Since the POE is basically a queueing system, it makes sense to use queueing theory models to describe it. In fact, queueing theory was developed in order to provide models that predict and to describe the behavior of systems that provide a service for randomly behaving demands. Pioneered first by the work of Erlang, “The Theory of Probabilities and Telephone Conversations” in 1909, and continued by Molina (1922) and others. They became the basis of queueing theory, and also the basis of the approximation for the general distribution which will be used in this section.

However, it is not always possible to develop an analytical model that accurately reflects the real world environment. In some cases, the modeler has to make as-sumptions to be able to develop a closed-form solution or even an approximation of the system. Tractable queueing models require reasonable analytical assumptions and are generally based on the “forgetfulness” aspect of the exponential distribu-tion. Simulation models offer a great deal of flexibility as they are generally able to describe with great detail almost any system. It can be used to validate models and approximations, and is “generally robust with respect to modeling distributional assumptions and allows for more realistic modeling of system interactions” (Curry and Feldman, 2011).

One server case

Fig. 6.2.: State Transition Diagram for M_t /E₂ /1 Illustration Case