The Jackson Networks As a Quite Good Solution For Analysing The Financial Analytical Service Facilities
Vladimir Simovic, Savo Vojnovic
University of Zagreb, Police College,
Avenija Gojka Suska 1, HR - 10000 Zagreb
The Republic of Croatia
Tel: +385 1 239 1341; Fax: +385 1 239 1415
E-mail:
Branko Kovacevic
University of Zagreb, Faculty of Economics
Kennedyev trg 6, HR - 10000 Zagreb
The Republic of Croatia
Tel: +385 1 239 1341; Fax: +385 1 239 1415
E-mail:
ABSTRACT:Objective of this work is to explain the modelling concept of the financial analytical function in the financial knowledge discovery model. We discuss the queuing networks with infinite queues in series and especially Jackson networks with m financial analytical service facilities as a quite good solution for analysing each service facility independently of other financial analytical service facilities. Also, in this work we discuss a conceptual solution for reducing the sum of costs (as a function of financial analytical service cost and cost of analytical waiting). To solve the large number of simulations we used the queuing M/M/s model with priorities that is based on a Poisson input (for exponential financial information inter-arrival time) and exponential output (for financial analytical service time), and which is mainly based on the birth-and-death process as a special type of continuous time Markov chain. An application of this model increases the group effectiveness, efficiency, and quality of the operational and strategic financial market investigative operations that are in usage during the whole financial knowledge discovery process.
Keywords:applied Jackson networks, financial analytical function
1INTRODUCTION
This work is a short explanation of the main Operations Research (OR) concepts and results, which are accomplished during the computing process of the modern financial analytical function (based on queuing networks), which is also a solid base for all simulation and modelling works in the future. Here are some remarks about “application of queuing theory and queuing networks”, with comments about financial analytical systems, based on “infinite queues in series and Jackson Networks". An illustrative example is presented at the end.
2THE APPLICATION OF QUEUING THEORY AND QUEUING NETWORKS
2.1 Analytical queuing theory
How does analytical queuing theory contribute important information required for achieving an economic (or other) balance between the cost of some financial analytical service and the cost associated with waiting for that financial analytical service? Indirectly, by predicting various characteristics of the analytical waiting line (such as the average waiting time, and similar). Providing too much financial analytical service involves excessive analytical costs. In the opposite situation we have almost the same problem. Not providing enough financial analytical service capacity causes the waiting line to become excessively long at times. This is costly in a sense, whether it is a financial or a social cost, the cost of lost financial analytical information, or the analytical cost of idle analytical employees and analytical servers. Because of that, analytical queuing theory is a specific concept, which involves the mathematical study of analytical queues and analytical waiting lines. The basic process assumed by this queuing model is the following (Figure 1). Customers are various data and information (in special analytical input forms) requiring (mainly) financial analytical service and generated over time, by an input source (financial events of the real world). These special analytical input forms (data and information about the financial world) enter the queuing system (analytical information queuing system of the whole financial analytical process) and join an analytical queue. At certain times, an analytical queue member is selected for analytical service by a rule known as the analytical queue discipline. The required analytical service is then performed for entered data and information (concerned financial business spheres) by the analytical service mechanism (of the whole financial analytical process), after which entered data and information leaves the analytical information queuing system.
Analytical information
Queuing systems
Input Analytical data Queue Service Served analytical
Sources and information System Mechanism data and information
Channels Facilities
Figure 1. The basic analytical information queuing systems
Financial events of the real world as the input source are usually assumed as an infinite (unlimited) rather then a finite (limited) population, because the input source size is the total number of distinct potential data and information that might require analytical service from time to time. The population, from which arrivals come, is referred to as the calling population. The number of data and information in the financial analytical information queuing system significantly affects the rate at which the input source generates new potential data and information. The assumption is that the statistical pattern by which the number of data and information are generated until any specific time (or over time) has a Poisson distribution. This case is the one where arrivals to the financial analytical information queuing system occur randomly, but at a certain fixed mean rate, regardless of how many data and information already are there (the size of the input source is infinite). The probability distribution of the time between consecutive arrivals is an exponential distribution, and is referred as interarrival time.
For this queuing model the queue is an infinite queue, and the analytical priority-discipline is based on analytical priority system. The analytical order in which members of the queue are selected for analytical service is based on their assigned priorities. Their assigned priorities are influenced with the financial analysis phase and with the mark of the financial analytical "data contents & information source evaluation system" (so called «4x4x2» evaluation system). The financial analytical "data contents & information source evaluation system" can be viewed like a conceptual tool for reducing the entropy of the modern financial analytical function. Modern financial analytical "data contents & information source evaluation system" («4x4x2» evaluation system) has very specific criterion, which in reality deals with minimally (442=) 32 linguistic variables for data contents and information source evaluation purposes. Assigned priorities of the analytical queue members are in fact something like: “1st class priority”, “2nd class priority”, “…”, and “N class priority”. The “1st class priority” has the highest priority and “N class priority” has the lowest. In other words, data and information are selected to begin analytical service in the order of their priority classes, but on a first-come-first-served basis within each priority class. A Poisson input process and exponential analytical service times are assumed for each priority class, with restrictive assumption that the analytical expected service time is the same for all priority classes, and the mean arrival rate differs among priority classes. Also, the rule is: once an analytical server has begun serving a potential analytical data (or information), the analytical service must be completed without interruption. This is referred to as a non pre-emptive analytical queuing model, where a potential analytical data (or information) being analytically served cannot be ejected back into the queue (pre-empted) if higher-priority class data (or information) enters the analytical queuing system. In fact the order in which a potential analytical data (or information) are analytically served is different from the “normal” or FIFO (First-come-In-First-served-Out) basis queuing order. The analytical service mechanism consists of one or more analytical service facilities, each of which has one or more parallel analytical service channels, so called analytical servers. For more than one analytical service facility, a potential analytical data (or information) may receive analytical service from a sequence of these analytical servers (service channels in series). A specific analytical queuing model specifies the arrangement of the analytical service facilities and the number of servers (parallel analytical service channels) at each one. Usually basic models assume one analytical service facility, with only one or a finite number of analytical servers.
2.2 Terminology and notation used by queuing theory
The following extension of standard terminology and notation was used:
state of analytical system = / number of a analytical data (or information) in queuing systemanalytical queue length = / number of a analytical data (or information) waiting for service (state of analytical system minus number of a analytical data being analytically served)
N(t) = / number of a analytical data (or information) in queuing system at time t (t)
Pn(t) = / probability of exactly n analytical data (or information) in queuing system at time t, given number at time 0
s = / number of analytical servers (parallel analytical service channels) in queuing
n = / mean arrival rate (expected number of arrivals per unit time) of new analytical data (or information) when n data (or information) are in the queuing system
n = / mean analytical service rate for overall financial analytical system (expected number of data (or information) completing analytical service per unit time) when n data (or information) are in the queuing system (Note: that is combined rate at which all busy analytical servers (serving data) achieve analytical service completions)
constant = / when the mean arrival rate n is a constant for all n
constant = / when the mean analytical service rate per busy analytical server is a constant, for all n1 (Note: in this case, n = s when n s , or when all analytical servers s are busy)
1/ = / expected interarrival time
1/ = / expected analytical service time
= / the utilization factor ( = /s) for the analytical service facility, or the expected fraction of time the individual analytical servers are busy
steady-state condition
of analytical system = / specific state of analytical system that is reached after sufficient time has elapsed (after transient condition of analytical system is finished), and when analytical system is essentially independent of the initial system state and elapsed time (where the probability distribution of the state of the analytical system remains the same)
Pn = / probability of exactly n potential analytical data (or information) in the analytical queuing system
L = / expected number of analytical data (or information) in the analytical queuing system
Ln= / expected analytical queue length (excludes analytical data or information being served)
W = / waiting time in analytical queuing system (includes analytical service time) for each individual analytical data or information
Wk= / steady-state or total expected waiting time in the whole analytical system (including analytical service time, or analytical supplying time), where W = E(W) and Wk is for a member of priority class k, which is k = 1, 2, ... , N
Wq = / waiting time in queue (excludes analytical service time) for each individual analytical data or information, where
Wq = E(Wq)
2.3 Mathematically based queuing model formulation
In relation to the proposed classification of the simulation models (see p. 14 in [12]) here we are dealing with abstract, dynamic, discrete, and stochastic models with numerical and analytical end solutions, which are primarily accomplished with a lot of discrete simulationsand with recording data on a developed model (for more details see pp. 16-21 in [12]). We are using a simplified explanation of the analytical function and basically complex multi-channel supplying system (for analytical information). For better clarity, suppose that in all financial analytical investigations we are analytically dealing mainly with financial based information and data. Also, we are dealing with a lot of “analytical information”, which are coming from legal sources (financial analytical process) and in some order that is proposed from well known “supplying theory” or “queuing theory” (see [12]). To solve the large number of the stochastic simulations of the modern financial analytical function we have used the queuing M/M/s model with non pre-emptive analytical priorities, which is based on a Poisson input (for exponential financial information interarrival time), and exponential output (for financial analytical service time) that is mainly based on the birth-and-death process (as a special type of continuous time Markovian chain). The queuing M/M/s model with non
pre-emptive analytical priorities assumes that both expected interarrival times (1/) and expected analytical service times (1/) have an exponential distribution, and that number of analytical servers is any positive integer (s). The benefit of this new simulation model of the financial analytical function is in the simple method (based mainly on statistical simulations) of measuring analytical capacity and capability of analysis, which is now in usage in the financial field (and partly in field of financial law). The financial analytical function of the financial analytical was prepared for investigations of various financial events, financial markets, subjects or entities, and for financial business operations control methods, etc. The model of modern financial analytical function (MFAF) has two basic analytical sub-systems: analytical service receivers (analytical function clients) and service suppliers (analytical function servers). Simplified, analytical service receivers (data or information) are coming in channels (with and without queues) with all kind of “analytical data or information” (maybe) interesting for financial analytical investigations (and especially for the financial analysis function). If there are free analytical service suppliers, then analytical service will be done in that moment. But, if there is no free analytical service supplier, service receivers are waiting for the service in queues. After the analytical service request was accomplished, analytical service receivers (fully analytical prepared information’s) are leaving the analytical system, or coming to the end of it (or to the specific destination point). The analytical supplying system can be simple or complex, open or closed, one or multi-channel, and also with and without priorities (see pp. 108-175 in [12]). Simplified, MFAF is a complex multi-channel analytical supplying system with priorities. It has minimally two or “n” analytical servers more then the classical financial analytical function, and it have a backward feedback sub-system (well known as «4x4x2» evaluation sub-system) [2]. MFAF is functionally complex, has multi-channel structure, backward feedbacks, tails with queuing and analytical information supplying sub-system with non pre-emptive priorities. That is reason why we are usually researching rather simplified (without backward feedbacks sub-system), but still complex multi-channel analytical supplying system with priorities.
Financial analytical service has added value as the result of OR based analysing and interpretation. It must be clear that the serious financial analytical practice is basically done with various experts, analysts and scientific financial analytical departments and that financial analytical resources are always finite. The operational financial analytical practice with finite number of financial analytical specialists and usually a lot of financial analytical cases during the same time period, produce a need for parallel and network working. The whole financial analytical process has almost clear heuristics, because intelligence is the resulting product from various systematically connected OR based processes, like: estimation, collection, evaluation, collation, integration, analysing and interpretation of data and information, development of hypotheses, dissemination of information, intelligence acting, co-ordination and automation (see [9-10]). During the research preserved we are not using the elementary modelM/M/1of the “queuing theory”, which is in fact one-channel (one server) analytical supplying system model with exponential distribution of inter-arrival times (of analytical information) and of (analytical supplying) service times. We are using the specific M/M/s model (for more details see pp. 628-755 in [6]), which assumes: that all inter-arrival times are independently and identically distributed, according to an exponential distribution (our input process is Poisson); that all analytical service times are independently and identically distributed according to another exponential distribution (our analytical service process is Poisson); and that the number of servers is s (any positive integer), but in the Croatian financial analytical practice and related analytical function they vary from a minimum 1 to maximum 7. With the equal distribution of analytical supplying time, with expected analytical service time about 1/ (n is mean analytical service rate for overall system, or expected number of clients (data or information) completing analytical service per unit time), and with exponentially distributed inter-arrival time of analytical information at expected average rate of 1/ (n is mean arrival rate, or expected number of arrivals per unit time), that is the most simplified type of Markovian analytical system with supposed infinite analytical capacity (Y = ), and with priorities in queue discipline (or without supposed FIFO queue discipline). We are researching analytical financial analytical cases in which there are no possibilities for any analytical closeness of multi-channels model of the analytical supplying function, or when the utilisation factor for the analytical service facility is s < 1 < s (because s = /s). For better clarity of the priorities concept, firstly we are talking about an elementary model of analytical supplying system M/M/1, which has a stationary state represented with these relations (see pp. 122-123 in [12]):
Pn = n (1-) ; = / < 1 ;Po = (1-) ; L = / (1-) = / ( - ) ;
Lq = 2 / (1-) = 2 / ( - ) ;
Wq = / ( - ) ;W = 1 / ( - ) ; Toccupied = 1 / ( - ) .
With introducing only two relative priority classes of the analytical supplying function in the same M/M/1 model, we have analytical information with higher priority of relative analytical supplying order, which have a mean arrival rate 1 (i priority class is equal 1), and analytical information with a lower priority of relative analytical supplying order, which have a mean arrival rate 2 (i priority class is equal 2). The parameter of their corporate (coupled) input exponential distribution is , and it is their arithmetic sum ( = 1 + 2). The analytical supplying function is the same for both types of analytical information, and has a mean service rate . Basic results for average measures of success (in stationary state condition) can be represented with these relations (see pp. 126-128 in [12]):