Deploying the Queueing Model in the Remote Patient Medical ...Queueing network models have been studied extensively [21], and there are numerous manufacturing applications of these

International Journals of Advanced Research in Computer Science and Software Engineering

ISSN: 2277-128X (Volume-7, Issue-6)

Research Article

June 2017

© www.ijarcsse.com, All Rights Reserved Page | 111

Deploying the Queueing Model in the Remote Patient

Medical Monitoring System Ifeoma V. Ngonadi

Dept. of General Studies (Maths& Computer Science Unit)

Petroleum Training Institute Effurun, Nigeria

DOI: 10.23956/ijarcsse/V7I6/01605

Abstract: A queueing system can be described as patients arriving for service, waiting for service if it is not immediate,

utilizing the service and leaving the system after being served. The queueing model is constructed so that queue

lengths and waiting time can be predicted.Remote patient monitoring enables the monitoring of patients’ vital signs

outside the conventional clinical settings which may increase access to care and decrease healthcare delivery costs.

This paper focuses on applying the queueing model to the remote patient medical monitoring system. This was

achieved by writing a java program which adapts an M/M/1queueing model that employs the first come first serve

discipline to store the patients’ medical records generated by a simulated mobile phone called the Intelligent Personal

Digital Assistant according to when they arrive at the queue. The queueing model gives the summary of the total

number of readings generated for each patient and also plots a graph so that the doctor can see the progression of

each patient’s readings at a glance. The result of this research work produces an enhanced remote patient medical

monitoring system which has the capacity to save lives.

Keywords: Queueing Model, Intelligent Personal Digital Assistant, Remote Medical Monitoring, Vital Signs, M/M/1

I. INTRODUCTION A queueing system can be described as patients arriving for service, waiting for service if it is not immediate, utilizing

the service and leaving the system after being served.

Queueing theory is the mathematical study of waiting lines, or queues[19].The queueing model is constructed so that

queue lengths and waiting time can be predicted [19]. Queueing theory is generally considered a branch of operations

research because the results are often used when making business decisions about the resources needed to provide a

service.

Queues form when entities that request service, typically referred to as customers, arrive at a service facility and cannot

be served immediately upon arrival. In healthcare delivery systems, patients are typically the customers and either

outpatient clinics or diagnostic imaging centers or hospitals are the service facilities.

Single queueing nodes are usually described using Kendall's notation in the form A/S/C where A describes the time

between arrivals to the queue, S the size of jobs and C the number of servers at the node [20][8]. Many theorems in

queueing theory can be proved by reducing queues to mathematical systems known as Markov chains, first described by

Andrey Markov in his 1906 paper [13].

The M/M/1 queue is a simple model where a single server serves jobs that arrive according to a Poisson process and have

exponentially distributed service requirements. In an M/G/1 queue the G stands for general and indicates an arbitrary

probability distribution. The M/G/1 model was solved by Felix Pollaczek in 1930 [17], a solution later recast in

probabilistic terms by AleksandrKhinchin and now known as the Pollaczek–Khinchine formula[10][23].

Remote medical monitoring of patients requires monitoring the physiological state of patients with acute or chronic

conditions or chronic disease states which predominantly derive decided prognosis advantages from intensive condition

tracking. More particularly, the invention is directed to a condition monitoring system which includes one or more

remote modular testing units and a central station. The remote units include physiological parameter testing modules to

acquire data from one or possibly many patients and communicate with a central station typically capable of interfacing

with a large number of patient-operated units or clinician-operated units testing many patients. The central station, in

turn, may interface and communicate with any number of other devices as by networking. Parameters checked may

include but are not limited to blood pressure, pulse rate, blood oxygen saturation, weight, blood glucose, temperature,

prothrombin (clotting) time and pulmonary function, including respiratory rate and depth. Other functions, such as ECG

(electrocardiograph) traces and infant breathing monitoring for detection of SIDS (sudden infant death syndrome) onset

are also contemplated [15].

A remote monitoring system operates with an instruction set to provide automated administration of health care to a

patient. In a preferred embodiment of the invention a central monitoring station receives data from a plurality of patients

connected with vital sign monitoring sensors which enter the queueing model from where the results are analysed and

kept in the server. The doctor can view the server from anywhere to see the results of the patients’ vital signs. A

particular patient’s graph can also be viewed by the doctor to see the progression of the patient’s vital signs readings.

Medical procedures are then administered to the patient and results taken as data. The data is made available to the

central monitor so that proper medical interpretation is enabled. A number of novel steps in the programming of the

dx.doi.org/10.23956/ijarcsse/V7I6/01605https://en.wiktionary.org/wiki/queuehttps://en.wiktionary.org/wiki/queuehttps://en.wiktionary.org/wiki/queuehttps://en.wikipedia.org/wiki/Queueing_theory#cite_note-sun-1https://en.wikipedia.org/wiki/Operations_researchhttps://en.wikipedia.org/wiki/Operations_researchhttps://en.wikipedia.org/wiki/Operations_researchhttps://en.wikipedia.org/wiki/Kendall%27s_notationhttps://en.wikipedia.org/wiki/Queueing_theory#cite_note-tijms-5https://en.wikipedia.org/wiki/Queueing_theory#cite_note-tijms-5https://en.wikipedia.org/wiki/Markov_chainhttps://en.wikipedia.org/wiki/Andrey_Markovhttps://en.wikipedia.org/wiki/Queueing_theory#cite_note-7https://en.wikipedia.org/wiki/M/M/1_queuehttps://en.wikipedia.org/wiki/Exponentially_distributedhttps://en.wikipedia.org/wiki/M/G/1_queuehttps://en.wikipedia.org/wiki/Probability_distributionhttps://en.wikipedia.org/wiki/Felix_Pollaczekhttps://en.wikipedia.org/wiki/Queueing_theory#cite_note-12https://en.wikipedia.org/wiki/Aleksandr_Khinchinhttps://en.wikipedia.org/wiki/Pollaczek%E2%80%93Khinchine_formulahttps://en.wikipedia.org/wiki/Pollaczek%E2%80%93Khinchine_formulahttps://en.wikipedia.org/wiki/Queueing_theory#cite_note-century1-13

Ngonadi International Journals of Advanced Research in Computer Science and Software Engineering



system are taken to assure that the right patient is being monitored, that the patient is being tested properly and that the

system is being monitored appropriately.

As the vital signs of patients are monitored from different locations and sent to the server in the hospital, it is necessary

to put these readings in a queue so that there will not be jamming of data. The M/M/1 queueing model which operates on

the First Come First Serve (FCFS) basis is used meaning that the packets of data enter the queue according to when they

arrive the queue. Java language is used to write a program for the implementation of this queuing model.

II. RELATED WORK Queueing theory has its origins in research by AgnerKrarupErlang when he created models to describe the Copenhagen

telephone exchange [19]. The ideas have since seen applications including telecommunication, traffic engineering,

computing[12] and, particularly in industrial engineering, in the design of factories, shops, offices and hospitals, as well

as in project management[18][14].

AgnerKrarupErlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper

on what would now be called queueing theory in 1909[1][2][4]. He modeled the number of telephone calls arriving at an

exchange by a Poisson process and solved the M/D/1 queue in 1917 and M/D/k queueing model in 1920[10]. In Kendall's

notation:

M stands for Markov or memoryless and means arrivals occur according to a Poisson process

D stands for deterministic and means jobs arriving at the queue require a fixed amount of service

k describes the number of servers at the queueing node (k = 1, 2,...). If there are more jobs at the node than there are

servers then jobs will queue and wait for service.

The queueing theory became an area of research interest to mathematicians after the 1940s[23]. In 1953 David George

Kendall solved the GI/M/k queue[9] and introduced the modern notation for queues, now known as Kendall's notation. In

1957 Pollaczek studied the GI/G/1 using an integral equation[16].John Kingman gave a formula for the mean waiting

time in a G/G/1 queue: Kingman's formula[11].

Queueing network models have been studied extensively [21], and there are numerous manufacturing applications of

these models [3]. Many of these models are not directly applicable to health systems design. Each model is specific to a

particular type of system (e.g., transfer lines with limited buffer) and typically requires either special techniques or

approximations to derive system performancemeasures. Therefore,we focus only on papers that utilize queueing network

methodology for modeling healthcare operations.

Whereas there are many attempts to represent networks of healthcare facilities as networks of queues, these networks are

typically not analyzed using queueingtheoretic approaches. Instead, a common approach is to use computer simulation to

obtain performance metrics of interest; [6] and [5]. Papers that use an analytic approach include [7] and [22]. Reference

[7] present a methodology for estimating expected utilization and service level for a class of capacity-constrained service

network facilities operating in a stochastic environment. In this work, the M/M/1 queuing model was used to gather the

patients’ medical data into packets and put on the queue. The queue functions according to first come first serve basis and

ensures that queues are not allowed to build up more than necessary by making sure that the waiting room size equals the

number of servers at each facility at all times.

III. SYSTEM ARCHITECTURE A. The Queueing Model for distribution of Patient data

Fig. 1: The Queueing Model Architecture

https://en.wikipedia.org/wiki/Agner_Krarup_Erlanghttps://en.wikipedia.org/wiki/Queueing_theory#cite_note-sun-1https://en.wikipedia.org/wiki/Telecommunicationhttps://en.wikipedia.org/wiki/Traffic_engineering_(transportation)https://en.wikipedia.org/wiki/Computinghttps://en.wikipedia.org/wiki/Computinghttps://en.wikipedia.org/wiki/Industrial_engineeringhttps://en.wikipedia.org/wiki/Queueing_theory#cite_note-3https://en.wikipedia.org/wiki/Queueing_theory#cite_note-3https://en.wikipedia.org/wiki/Agner_Krarup_Erlanghttps://en.wikipedia.org/wiki/Queueing_theory#cite_note-8https://en.wikipedia.org/wiki/Queueing_theory#cite_note-8https://en.wikipedia.org/wiki/Queueing_theory#cite_note-10https://en.wikipedia.org/wiki/Poisson_processhttps://en.wikipedia.org/wiki/M/D/1_queuehttps://en.wikipedia.org/wiki/M/D/k_queuehttps://en.wikipedia.org/wiki/Queueing_theory#cite_note-century-11https://en.wikipedia.org/wiki/Queueing_theory#cite_note-century1-13https://en.wikipedia.org/wiki/David_George_Kendallhttps://en.wikipedia.org/wiki/David_George_Kendallhttps://en.wikipedia.org/wiki/David_George_Kendallhttps://en.wikipedia.org/wiki/Queueing_theory#cite_note-14https://en.wikipedia.org/wiki/Kendall%27s_notationhttps://en.wikipedia.org/wiki/Integral_equationhttps://en.wikipedia.org/wiki/Integral_equationhttps://en.wikipedia.org/wiki/Integral_equationhttps://en.wikipedia.org/wiki/John_Kingmanhttps://en.wikipedia.org/wiki/Mean_sojourn_timehttps://en.wikipedia.org/wiki/Mean_sojourn_timehttps://en.wikipedia.org/wiki/Mean_sojourn_timehttps://en.wikipedia.org/wiki/G/G/1_queuehttps://en.wikipedia.org/wiki/Kingman%27s_formulahttps://en.wikipedia.org/wiki/Kingman%27s_formulahttps://en.wikipedia.org/wiki/Kingman%27s_formula




PID 1 Patient ID 1: This can be from Pid 1 to Pid n.

SP 1 Specialist 1: This can also be form Sp 1 to Sp n.

Fig. 2: Block diagram of the queuing model

The M/M/1 queue consists of a server which provides service for the packets of data from the patients who arrive at the

system and depart. It is a single-server queuing system with exponential interarrival times, exponential service times and

first-in-first-out queue discipline. If a packet of data from a patient arrives when the server is busy, it joins the queue (the

waiting line). There are two types of events: arrival events (A) and departure events (D). The following quantities are

used in representing the model:

AT = arrival time

DT = depature time

Fig. 3: Flowchart of the queue simulation




Fig. 4: The first few events in simulation

In the first few events in simulation, the packets of data from the patient arrive the queueing model as the first arrival in

time 0 and the interarrival time which is the time between the first arrival and the next arrival is denoted by T0 while the

next arrival is denoted by T1. The first packets of data to arrive the queue is also the first to depart from the queue which

is the first departure.

In the queueing model, vital signs are collected by the sensor on the patients’ body, sent through the Bluetooth (This data

is simulated in the IPDA) to the IPDA which transmits this data by WiFi to the router which further transmits the data

wirelessly to the hub which is in the server.

The hub acts as a data repository where these data are stored and sent to the doctor when there is an abnormal situation.

B. The Queueing Model Queueing models can be represented using Kendall’s notation

A/B/S/K/N/D

Where

A is the interarrival time distribution

B is the service time distribution

S is the number of servers

K is the system capacity

N is the calling population

D is the service discipline assumed

1) The Arrival Rate The data arrive as packets of data from different patients wearing the sensors into the hub.

Let Ci be the interarrival time between the arrivals of the (i – 1)th and the ith patients, the mean(or expected) inter-arrival

time is denoted by E(C) and is called β; = 1/E(C) the arrival frequency. The data is simulated in the IPDA and the

interarrival time is 25 seconds and therefore β = 1

25 = 0.4.

2) Service Mechanism This is specified by the number of servers (denoted by s) each server having its own queue or a common queue and the

probability distribution of the patient’s service time.

Let Si be the service time of the ith patient, the mean service time of a customer is denoted by E(S) = 𝜇 =1

𝐸 𝑆 the service

rate of a server. The service time here is 10 seconds and therefore µ = 1

10 = 0.1.

3) Queue Discipline Discipline of a queuing system means the rule that a server uses to choose the next patient from the queue (if any) when

the server completes the service of the current patient.

The queue discipline for this system is

Single Server- (FIFO) First In First Out i.e. patients data are worked on according to when they came to the queue.

4) Measures of Performance for the Queuing System Let

Di be the delay in queue of the ith patient

0

Interarrival time

T0

First arrival Next arrival

T1

Simulated time

Scheduled departure

time First departure




Wi be the waiting time in the system of the ith patient

F(t) be the number of patients in queue at time t

G(t) be the number of patients in the system at time t = F(t) + No of patients served at t.

Then the measures,

D = lim𝑛→∞ 𝐷𝑖𝑖=𝑛1=1

𝑛 and (1)

W =lim𝑛→∞ 𝑊𝑖𝑖=𝑛1=1

𝑛 (2)

are called the steady state average delay and the steady state average waiting time in the system.

Also the measures,

F =lim𝑛→∞1

𝑇 𝐹 𝑡 .𝑑𝑡

𝑇

0 and (3)

G = lim𝑛→∞1

𝑇 𝐺 𝑡 .𝑑𝑡

𝑇

0 (4)

are called the steady state time average number in queue and the steady statetime average number in the system.

5) Single Channel Queue [M/M/1] : {FCFS or FIFO} Queue System

6) Arrival Time Distribution This model assumes that the number of arrivals occurring within a given interval of time t, follows a poisson distribution

with parameter 𝛽 𝑡. This parameter 𝛽 𝑡 is the average number of arrivals in time t which is also the variance of the distribution. If n denotes the number of arrivals within a time interval t, then the probability function p(n) is given by

P(n) = (𝛽)𝑛

𝑛!𝑒−𝑥 n = 0,1,2…. (5)

The arrival process is called poisson input

The probability of no(zero) arrival in the interval [0,t] is,

Pr (zero arrival in [0,t]) = 𝑒−𝛽𝑡 = p(0)

Also

P(zero arrival in [0,t]) = P(next arrival occurs after t)

= P(time between two successive arrivals exceeds t)

Therefore the probability density function of the interarrival times is given by,

𝑒−𝛽𝑡 for t > 0 This is called the negative exponential distribution with parameter 𝛽 or simply exponential distribution. The mean inter-arrival time and standard deviation of this distribution are both 1/(𝛽) where, (𝛽)is the arrival time.

7) Performance Measures The average number of units in the system G can be found from

G = sum of [n*Pn] for n = 1 to ∞

G = 𝛽

𝜇−𝛽 =

𝑃

1−𝑃 where P =

𝛽

𝜇 (6)

The average number in the queue is

F = (G – (1 – P0) (7)

Sum of [(n-1)*Pn] for n = 1 to ∞

F = 𝛽2

𝜇 (𝜇−𝛽) =

𝑃2

(1−𝑃) (8)

The average waiting time in the system (time in the system) can be obtained from

W = 𝐺

𝛽 =

1

𝜇−𝛽 and (9)

D = W -1

𝜇=

𝛽

𝜇 .(𝜇−𝛽). (10)

The traffic intensity P (sometimes called occupancy) is defined as the average arrival rate (lambda) divided by the

average service rate (mu). P is the probability that the server is busy.

P = 𝛽

𝜇 (11)

The mean number of customers in the system (N) can be found using the following equation:

(12)

You can see from the above equation that as p approaches 1 number of customers would become very large. This can be

easily justified intuitively. p will approach 1 when the average arrival rate starts approaching the average service rate. In

this situation, the server would always be busy hence leading to a queue build up (large N).

Lastly we obtain the total waiting time (including the service time):

T = 1

𝜇−𝛽 (13)




IV. RESULTS A program was written in java to simulate the remote patient monitoring system. A simulated mobile phone called the

Intelligent Personal Digital Assistant (IPDA) was used for the simulation. It generates the patients’ vital signs which are

sent to the queueing model. These readings are generated every twenty five seconds. The queueing model stores these

records according to when they arrive. A graph can also be plotted using the readings for each patient so that the doctor

can view each patients’ readings at a glance and proffer his solution.The results of these simulations are shown in the

figures below:

Fig. 5: The IPDA Fig. 6: The simulated reading for a patient

Fig. 7: The detailed readings of a patient




Fig. 8: The Mobile Queue

Fig. 9: The graph for a patient showing the readings of the vital signs

V. CONCLUSION The remote patient monitoring system has been discussed. It has been shown that a simulated mobile phone called the

IPDA can be used to simulate the readings of the vital signs of patients. These readings are generated every twenty five

seconds called the arrival time and continues as long as the application is on.




The queueing model was adapted and incorporated into the work to store these readings of the vital signs of patients. The

queueing model called the mobile queue in this work gives a summary of the total number of readings generated for each

patient, the time of arrival, the number of channels, the service time, the service rate, the mean patient arriving the system

and the traffic intensity.

It is shown also that the queueing model enhances the remote patient medical monitoring system by giving the doctor the

detailed readings of each patient’s medical records at a glance. This aids optimum performance of the system and saves

lives.

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Documents

Deploying the Queueing Model in the Remote Patient Medical ...Queueing network models have been studied extensively [21], and there are numerous manufacturing applications of these