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A Queue Network M/M/1/^ Model
Rajitha C., Chacko V. M.
Department of statistics, St. Thomas College (Autonomous) Thrissur-1, Kerala, India chackovm@gmail.com
Abstract
In this paper we model an open queueing network of cardiac treatment section in medical sector. Assume arrival of patients follows Poisson and service times at stations have exponential distribution. The performance measures of the system are evaluated. The steady state characteristics of the network are obtained and each station solved independently by using M/M/1/^ model while blocking and non-blocking exists. Blocking occurred when at least one service center has limited queueing space or capacity before it. An illustrative example is given.
Keywords: Queueing networks, Blocking, Steady state characteristics.
I. Introduction
Collection of interactive queueing systems is known as network of queues. Queueing networks mainly classified as open queueing networks, closed queueing networks and mixed queueing networks. Open queueing networks described as customers can arrive from outside the system at any node and depart from the system from any node. At least one service center has limited waiting space or capacity, which are classified in to restricted queueing networks. Blocking may arise in a network of queues where some or all queues have finite buffer capacity [2]. Since there is restriction in waiting space between the stations, there may occur blocks.
Many relevant studies on open restricted queue systems are done by Hunt [3], Takahashi et al.[8], Perros and Atlok [6], Koizumi et al.[5], Sreekala and Manoharan [7] and Arum Helmi Manggala Putri et al [1]. Hunt [3] used a sequential series model to obtain solution for a two station series queue with limited waiting space between stations. An approximate analysis for open queueing networks with blocking done by Takahashi et al.[8] and Perros and Atlok [6]. Koizumi et al [5] analysed blocking in open restricted queueing system by decomposition method. Recently analysis of restricted queueing networks-a blocking approach with special reference to health care system studied by Sreekala and Manoharan [7].
In this paper first we study an open queueing network of Cardiac section with infinite capacity in each station. Steady state equations and performance parameters are obtained. A brief description of the model is done in section II. Diagrammatic representation and congestion types are given in Section III. Derivation of steady state equations in without blocking and analysis of each station using decomposition approach in with blocking are given in section III. Numerical
Rajitha C., Chacko V. M.
A QUEUE NETWORK M/M/l/™ MODEL
analysis is section IV. Conclusions are given in last section.
II. Model description
We can consider Out Patient (OP) section of Cardiac treatment in government medical college, as an example of open queueing network. There are five stations are defined in this queueing network. In first node Sx gives token for every customer arriving to the hospital, customers arrive according to homogeneous Poisson process. Second station stands for pressure checking which follows M/M/1/^/FCFS schedule. Doctors are available in the third and fourth node and these nodes considered as a single node. S3 and S4 also follow M/M/1/K/FCFS. Fifth node is for treatment. After the diagnosis, some patients in third and fourth node leave from the system with probability a3 and a4and remaining patients admit for treatment with probability 1-a3 and 1-a4.There are some situations where usual admission procedures cannot follow. Example: - accident cases or heart attack.
Methodological Framework
Consider an open queueing network of OP section of Cardiac treatment in medical college with five single servers. LetSj ( i = 1,2,... 5 ) denote stations. Arrival pattern of customers to the system according to homogeneous Poisson process with rate X. Service times are exponentially distributed with rate ^ (i = 1,2,... 5). Queue discipline is FCFS basis. Waiting space S5 and between stations one and two are of infinite capacity and other stations are finite. Therefore blocking happens only between S3 — S5 andS4 — S5. In this paper, model the flows S2 —^ S3, S2 —^ S4, S3 —^ S5 and S4 — S5. Arrival to each node is according to Poisson process. Diagrammatic representation of the model is given in figure 1.
Finite
Infinite
Infinite
l-or.
«3
Figure 1. OP section of Cardiac treatment asqueue network with blocking
Since waiting space between stations three, four and five are of finite capacity there may arise blocking between S3 — S5 and S4 — S5. We model the flows S3 — S4 and S4 — S5. Arrival to each node is according to Poisson process. The types of congestion listed in table 1.
Table.1.Congestion types
Flow Cause of congestion Facing station Congestion type
Si ^ S2 Not applicable Not applicable No congestion
S 2 ^ S3 S3 is full Classic Congestion
S2 ^ S4 S4 is full S2 Classic Congestion
53 55 is full S3 Blocking
S4 S5 is full s4 Blocking
III. Steady -State Analysis
In this section we first assume that every station has infinite waiting space and analyze stations without blocking. The steady state analysis of some related models can be seen in Gross and Harris [3] and Bose [2].
Steady State Analysis without blocking
The routing probability matrix generally defined as
p =
»00 r01 r02 r03 r04 »05 »06
r10 r11 »12 »13 »"14 »"15 »"16
r20 »"21 r22 r23 »24 »25 »26
r30 »"31 r32 r33 »34 »35 »"36
r40 »"42 »43 »44 »"45 »46
»50 »51 »52 »53 »54 »55 »56
-r60 »61 »62 »63 »64 »65 »66
where rij is the routing probability from station i to station j (i,j= 1,2,..,6). The routing probability matrix of our model based on figure 1 is
P =
0 1
0
1 — % 0 0 0 0
0 0
«2 0 0 0
0 0
1 — «2 1 0 0
0 0 0
1 —«3
1 — a4 0
We can find ^ : i = 1,2,... 5 (Total arrival rates) by solving the traffic equations:
h= X
X2= (1 — aJX X3 =a2(1 — «i)X X4 =(1 — a2)(1 — a1)X
X5 =(1 — «2X1 — «i)X[(1 — «3) + (1 — «4)].
We assume there is an infinite buffer between stations. So we can solve each station independently applying M/ M/ 1/ro queueing model.
Average queue length and Average queue delay
Average queue length of station i is obtained from the formula [3].
=
Pi
1—Pi
where p; = Xj/uj <1 (i=1,2,.. .,4) and p; = —>1 (i=5), in the case of non-steady state,
Mi
model cannot be applicable to the queueing system.
By using Little's formula[3] we can obtain the average steady state waiting time,
=
-;(i-p;)
,i = 1,2,. 5.
(1)
analytical (2)
i
3
a
4
1
2
p
Steady state analysis with blocking
Blocking exists between stations when some stations are finite and congestion at any particular station could potentially affect congestion levels at all upstream stations. In order to find interactions between stations, modified Jacksons approach can be used with the help of effective
service time by Takahashi et al. [9]. Here we assume effective service times follow exponential
1
distribution. The mean effective service time at station i is denoted by —. Effective waiting time is defined as the convex combination of waiting times.
1 = ri0 (1)+Trij(1 + W]), Hi W ¿-r> v
J
where ri0 is the routing probability of patients leaving from state i without facing any wait,r^ is the routing probability from station i to j. In our model stations S3 and, S4 face blocking. The effective service time corresponding to S3 and, S4 are
(3)
(4)
Using equations (1) and (2), we obtain steady state queue lengths and waiting times in terms of effective service times.
Analysis of Stations
By using single node decomposition approximation by Takahashi et al. [9] the steady state of every station can solve independently from last station to first station. The steady state of each finite station (M/M/1/ro queue) is analyzed using this approximation.
Analysis of station six (S5)
In our model, station five represents who needs recheck and treatment in the clinic. The downstream node S3 and S4 are finite. S5 face blocking if S3 is full. Corresponding to S5 the queue length and queue delay obtained by solving (1) and (2) in terms of effective service times (3). The queue length corresponding to S5 is
L35 = Lq5(X35 /w
= Lqq(T3qX3/Xq)
whereL^ is thequeue length of blocked persons at S3 waiting to enter S5. S5 face blocking if S4 is full. Corresponding to S5the queue length and queue delay obtained by solving (1) and (2) in terms of effective service times (4). The queue length corresponding to S5 is
L\q = Li*(X4q/'Xq) = Lqq(T4qX4/Xq)
where L45 is the queue length of blocked persons at S4 waiting to enter S5. This method is applicable only when traffic intensity less than one.
Analysis of station four (S4)
Station four represents patients who entered for doctors checking. The downstream node S2 is also infinite. The queue length corresponding to S5 is
Lq = Lq
where is the queue length of blocked persons at S2 waiting to enter S4.
Analysis of station three (S3)
Station three represents patients who entered for doctors checking. The downstream node S2 is also infinite. The queue length corresponding to S3 is
^23 =
where Lq23 is thequeue length of blocked persons at S2 waiting to enter S3.
Analysis of station two (S2)
Station two represents patients who entered for pressure checking. The downstream node S1 is also infinite. The queue length corresponding to S2is
^23 = ^3'
IV. Numerical Analysis
Data taken from Cardiac Section of a Medical College. To find out the Performance
Parameters of with blocking and without blocking, the statistical analysis is conducted. A = 8.48,
1 111 a1 =0.85, a2 = 0.45, a3 = 0.98, a4 = 0.97, —=0.29, — = 0.91, — = 0.83, — = 0.75. Performance
1 2 3 4 Mi M2 M3 M4
parameters are computed and given in table 2.
Table 2. Performance Parameters
Station Performance With Blocking Without Blocking
Parameters
S2 1.21 1.21
w* 2.08 2.08
S3 0.54 0.54
w* 0.91 0.91
S4 0.66 0.66
< 0.75 0.75
The congestion rate and waiting time of S2 is high compared to other two stations. Blocking exists in station 5 but the traffic intensity of station 5 is greater than one. Steady state doesn't exist for this station. Normal methods cannot applicable for analyzing non-steady state queue system. Bounding of capacity of queue, Monte Carlo simulation and increase servers to the system are the possible methods for solving non-steady state queues.
V. Conclusion
We studied the Cardiac section of a Medical College as an open restricted queueing network. Blocking exists due to 5th node. Steady state equations are obtained without blocking and with blocking cases assuming traffic intensity less than one. Node to node decomposition method is used to get performance measures. But the traffic intensity of 5 th node greater than one. We cannot analyze through steady state equations. To find performance measures we can use any of these methods-Bounding of capacity of queue, Monte Carlo simulation and increase servers to the system.
References
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[2] S.K. Bose. An introduction to queueing Systems, Kluwer academic/ Plenum publishers, New York, 2002.
[3] G. Gross and C. Harris. Fundamentals of queueing theory, John Wiley and sons, 1998
[4] G. C. Hunt. (1956). Sequential arrays of waiting lines. Operations Research, 4:674-683.
[5] J.R. Jackson. (1957). Network of waiting lines. Operations Research, 5: 518-521.
[6] N. Koizumi, E. Kuno and T. E. Smith. (2005). Modeling patients flows using a queueing network with blocking. Health care Manag. Sci, 8:49-60.
[7] A. J. H. G. Perros and T. Attiok.(1986). Approximate analysis of open networks of queues with blocking: Tandem Configurations. IEEE Trans. Soft. Eng. 12:450-461.
[8] M. S. Sreekala and M. Manoharan.(2016). Analysis of restricted queueing networks- A blocking approach. Journal of Statistical Science and Application, 2: 220-230.
[9] Y. Takahashi, H. Miyahara and T. Hasegawa. (1980). An approximation method for open restricted queueing Networks. Operations Research, 28:594-602.
