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We consider a real time data acquisition and processing multiserver system with identical servers (such as unmanned aerial vehicles, machine controllers, overhearing devices, medical monitoring devices, etc.) which can be maintained/programmed for different kinds of activities (e.g. passive or active). This system provides a service for real time tasks arriving via several channels (such as surveillance regions, assembly lines, communication channels, etc.) and involves maintenance. We focus on the worst case analysis of the system with ample maintenance facilities exponentially distributed time to failure and maintenance times. We consider two kinds of models (with and without nonpreemptive priorities) and provide balance equations for steady state probabilities and various performance measures, when both operation and maintenance times are exponentially distributed.

Real time systems (RTS) are imbedded in most modern technological structures, such as production control systems, robotic and telecommunications systems, radars, self-guided missiles, reconnaissance, aircraft and space stations, etc. These systems are widely used for the monitoring and control of physical processes.

The sustained demands of the environments in which RTS operate pose relatively rigid requirements (usually stated as time constraints) on their performance.

In RTS, a calculation that uses temporally invalid data or an action performed too early/late may be useless, and sometimes harmful―even if such a calculation or action is functionally correct. These systems are often associated with applications, in which human lives or expensive machinery may be at stake.

There exists a rich literature covering RTS and different scientific communities are treating various problems in this area. We will site only a small portion of it.

Liu and Layland in their famous work [

Dhall and Liu [

Several authors treated real-time Flexible Manufacturing Systems (FMS). Tawegoum et al. [

Efficient metaheuristic methods, such as Tabu Search (Glover and Laguna [

We will focus on RTS with a zero deadline for the beginning of job processing. Queueing of jobs in such systems is impossible, since jobs are executed immediately upon arrival, conditional on system availability.

That part of the job which is not processed immediately is lost forever and cannot be served later.

The following works treated this kind of RTS. Kreimer and Mehrez ( [

The work presented here is a generalization of results obtained by Shimshi and Kreimer [

The paper is organized as follows: In Section 2, the description of the model is presented. Section 3 provides balance equations for RTS with nonpreemptive priorities. Section 4 provides balance equations for RTS without priorities. Section 5 is devoted to computation of various performance characteristics. Finally, Section 6 contains conclusions.

The most important characteristics of RTS with a zero deadline for the beginning of job processing are summarized in Kreimer and Mehrez [

1) Jobs/data acquisition and execution are as fast as the data arrival rate.

2) Jobs are executed immediately upon arrival, conditional on system availability. Between jobs arrival and their execution, delays are negligible.

3) That part of the job which is not executed in real time is lost forever. Storage of non-completed jobs or their parts is impossible. Thus, queues of jobs do not exist in such an RTS.

Nevertheless, some important elements of queueing theory can be successfully applied in analysis of these RTS.

We consider a multiserver multichannel RTS consisting of N identical servers (e.g. unmanned aerial vehicles- UAV, machine controllers, overhearing devices, medical monitoring devices, etc.) that provide service for the requests of real time jobs, arriving via r identical channels (e.g. surveillance regions, assembly lines, communication channels, hospital patients, etc.).

Each server can serve different channels, but not simultaneously. There is exactly one request in each channel at any instant (there are no additional job arrivals, while the channel is busy), and therefore one server at most is used (with others being on stand-by or in maintenance or providing the service to another channel) to process the job of this channel at any given time. Thus, the total number of jobs in the system is r (is equal to a number of channels). Different parts of the same job can be served by different servers. Any part of the job that is not processed immediately is lost.

It is assumed that there are ample identical maintenance teams available to repair (with repair times

The system works under a maximum load (worst case) of nonstop data arrival, which is equivalent to the case of a unique job of infinite duration in each one of r channels. This kind of operation is typical in high performance data acquisition and control systems. Decisions based on the nonstop data flow arriving via all r channels must be made extremely fast―in real time. If, during some period of time of length T, there is no available server to serve one of the channels, it means that the part of the job of length T is lost forever.

Our purpose is to provide balance equations for steady-state probabilities of this system, its availability and other performance characteristics. We will consider two policies/models: 1) nonpreemptive priority; 2) random choice of the fixed server.

First we will provide the list of various terms, which are used in our equations.

We suppose that servers of first activity type have the highest priority, servers of the second activity type are after them, and so on, finally, servers of the m-th activity type have the lowest priority (

We denote

Theorem 1. Steady state probabilities for the model with nonpreemptive priorities can be found from the fol- lowing system of balance equations:

where

And

where

Proof:

1) First we consider the case

for the state

for the states

for the states

for the states

for the states

Now Equation (1) follows from Equations (3)-(7).

2) Next, we consider the case

Equation (3) again for the state

Equation (4) for the states

Equation (6) for the states

for the states

for the states

for the states

for the states

for the states

for the states

for the states

for the states

Now, Equation (1) follows from Equations (3), (4), (6), (8)-(15).

Q.E.D.

We suppose that when the operating server must go to maintenance, the server which takes its place is chosen among the fixed servers in a random way (

Theorem 2. Steady state probabilities for the model without priorities can be found from the following system of balance equations:

where

And

where

Proof:

1) First, we consider the case

2) Next, we consider the case

Equation (3) for the state

Equation (4) for the states

Equation (6) for the states

Equation (8) for the states

Equation (9) for the states

Equation (14) for the states

And Equation (15) for the states

For the states below, we will have following new equations:

for the states

for the states

for the states

for the states

Now, Equations (16) follows from Equations (3), (4), (6), (8), (9), (14), (15), (18)-(21).

Q.E.D.

In this section, we shall show how to compute some useful performance characteristics of the RTS under consideration.

Each server can be in one of following positions:

a) Busy (operating);

b) Out of order (in maintenance);

c) On stand-by;

Each channel can be in one of two positions:

a) In service;

b) Out of service.

Keeping in mind that, the state of the system is represented by the vector

Number of fixed servers is

Number of operating fixed servers (the number of channels in service) is

Number of fixed servers of i-th activity type being on stand-by is

Number of all fixed servers being on stand-by is

Number of broken servers is

Number of channels out of service is

Now we can obtain corresponding average values, using steady state probabilities, namely:

The use of RTS relies on the principle of availability, which can be given in our case by the following formula

Another important performance measure is the average cost of system operation during time unit. Denote

In this paper, we provided equations for steady-state probabilities, as well as formulas for availability, average cost function and other performance characteristics of multiserver and multichannel RTS with different activity types and ample maintenance facilities. We have examined models with nonpreemptive priorities and without them. Practitioners and researchers can submit their parameters in equations for steady state probabilities and performance characteristics and to make their choice between two models in order to solve real life problems. Further research should address following problems:

The use of queueing theory methodology will have significant benefits in analysis of these systems.

JosephKreimer,EdwardIanovsky, (2015) Real Time Systems with Nonpreemptive Priorities and Ample Maintenance Facilities. Journal of Computer and Communications,03,32-45. doi: 10.4236/jcc.2015.37004

N number of servers

r number of channels

m number of server’s activity or quality kinds/types

D penalty cost for the time unit during which one of channels was not served

Av system availability

TC an average total cost of system operation per time unit