CC BY
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Discrete & Continuous Models
#& Applied Computational Science 2022, 30 (2) 160-182
ISSN 2658-7149 (online), 2658-4670 (print) http://journals-rudn-ru/miph
Research article
UDC 519.872:519.217
PACS 07.05.Tp, 02.60.Pn, 02.70.Bf
DOI: 10.22363/2658-4670-2022-30-2-160-182
Analysis of queuing systems with threshold renovation mechanism and inverse service discipline
Ivan S. Zaryadov1,2, Hilquias C. C. Viana1, Tatiana A. Milovanova1
1 Peoples' Friendship University of Russia (RUDN University), 6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation 2 Institute of Informatics Problems, FRC CSC RAS, 44-2, Vavilova St., Moscow 119333, Russian Federation
(received: February 22, 2022; revised: April 18, 2022; accepted: April 19, 2022)
Abstract. The paper presents a study of three queuing systems with a threshold renovation mechanism and an inverse service discipline. In the model of the first type, the threshold value is only responsible for activating the renovation mechanism (the mechanism for probabilistic reset of claims). In the second model, the threshold value not only turns on the renovation mechanism, but also determines the boundaries of the area in the queue from which claims that have entered the system cannot be dropped. In the model of the third type (generalizing the previous two models), two threshold values are used: one to activate the mechanism for dropping requests, the second — to set a safe zone in the queue. Based on the results obtained earlier, the main time-probabilistic characteristics of these models are presented. With the help of simulation modeling, the analysis and comparison of the behavior of the considered models were carried out.
Key words and phrases: queuing system, active queue management, renovation mechanism, threshold, time-probabilistic characteristics, GPSS modelling
1. Introduction
According to [1] the problem of congestion avoidance for communication networks does not have a satisfying solution, so the development and the analysis of new active queue management (AQM) algorithms appears to be the actual task for researches [2]-[13] and practitioners [14]-[24].
In this paper we will consider queuing systems with probabilistic renovation mechanism, which allows to adjust the number of packets in the system by dropping (resetting) them from the queue depending on the ratio of a certain control parameter with specified thresholds [25], [26] at the moment of the end of service on the device (server) [27]-[29] in contrast to standard RED algorithm, when a possible reset occurs at the time of the next packet arrival and the control parameter is an exponentially weighted average queue
© Zaryadov I.S., Viana H. C. C., Milovanova T.A., 2022
This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/
length [30]-[34]. In our models the renovation mechanism uses one or two thresholds (which determine as the place in the buffer from which the packets are dropped, but also the place to which the reset of packets occurs).
The previous works devoted to the analysis of queuing systems with threshold based renovation are [35]-[38]. In [35], [36] some aspects of using the renovation mechanism (different types of renovation, definitions and brief overview were also given) with one or several thresholds as the mathematical models of active queue management mechanisms were considered. Some results of comparing classic RED algorithm with renovation mechanism were presented. In [37] two queuing models with threshold based renovation mechanism were presented: in the first model the threshold value is only responsible for activating the renovation mechanism (the mechanism for probabilistic reset of claims), in the second model the threshold value not only turns on the renovation mechanism, but also determines the boundaries of the area in the queue from which claims that have entered the system cannot be dropped. In [38] the queuing system with two threshold values (one to activate the mechanism for dropping requests, the second — to set a safe zone in the queue) for renovation mechanism was investigated. All three queuing systems have been studied for the service discipline FCFS (First Come First Served), and in this article we will present some results for the discipline LCFS (Last Come First Served). The study will again be carried out using embedded Markov chains. We will not consider in detail the derivation of the stationary distribution of the number of customers (which does not depend on the service discipline and presented in [37], [38]) and will focus only on the service (reset) probabilities and on time characteristics.
The structure of the article is following. In the section 2 the results for the queuing model, where the threshold value is only responsible for activating the renovation mechanism, are presented; the section 3 is devoted to the queuing model, in which the threshold value not only turns on the renovation mechanism, but also determines the boundaries of the area in the queue from which claims that have entered the system cannot be dropped. In section 4 the characteristics for the queuing system with two threshold values (one to activate the mechanism for dropping requests, the second — to set a safe zone in the queue) for renovation mechanism are presented. In section 5 the results of GPSS simulation are considered. The last section 6 concludes the paper with the short discussion.
2. The first model
Consider the GI/M/ 1/ro queuing system, shown in the figure 1, with the implemented renovation mechanism and a threshold value Q1, which determines the boundary in the queue, starting from which the dropping of customers begins. If the current number of packets in the system i < Q1 + 1 (the threshold value Q1 is not been overcome), then none of the packets will be dropped from the queue. If the current number of packets in the system i ^ Q1 + 1, then with probability q the packet finishing the service can drop all packets from the queue and leave the system, or with probability p = 1 — q the serviced packet simply leaves the system.
Figure 1. Queuing system model
2.1. The service probability and the loss probability for a received
packet
Let p(loss) be the probability that the packet received in the system will be dropped by renovation mechanism and let p(loss) be the probability that a packet arriving and finding in the system exactly i packets will be dropped.
Let p(loss) (x) be the probability that in a time less than x a packet that finds other i packets in the system will be dropped. Then:
to
(loss) / (loss) /
Pi = I Pz,0 (x)dx, 0
where p(1°ss)(x) is the probability that in time less than x the packet, before which there are i other packets in the queue and after which there are other j packets, will be dropped, i,j > 0.
Let t;(1°ss) (x) be the probability density functions and let pf°ss) (s) be the Laplace-Stieltjes transforms. Then:
TO
(loss) / \ / (loss)/ (loss) / \ / (loss)/ \ J
Tz,j )(x) = (Pi,j ) (x))x , Pz,j ) (s)= I Tz,j ) (x)dx.
0
a) If i + j + 1 < Q1 the threshold is not crossed, then:
X J k
0 fc=o k-b.1) If i + j+1>Q!, i + 1 , then:
r(joss) (x) = V JT^e-^ ■ pk-1 ■ q ■ A(x)+
1 k= 1 (k-1)-
y ' k
+ I e-"y ■pmin(k,i+1+J-Qi) dA(y)Tf°ss) (x-v).
b.2) If i + j+1>Q1, i + 1> Q1, then:
rSss) (x) = t ■pk-i ■q ■ J{x)+
y
+ J t ^¡f e-"y ■pk^^(o-i+i (x — y)■ 0 k=0 '
Then for the Laplace-Stieltjes transforms p(l°ss) (s) we have:
a) If i + j + 1 < Q1, then:
piTH*) = 1*{k)(p + °) ■ p^+1 (*)■
k=0 '
b.1) If i + j+1>Q1, i + 1 , then:
min(j,z+1+j-Qi) k_1 k
pif (s) = i Lw—rt^(k-1) + s)■pk-1 ■ 9+
+ ^t ^Pmin(k'z+'+1-Qi>a(k) fa + s) ■ pi°s{+1 (s).
k=o k' '
b.2) If i + j+1>Q1 ,i + 1>Q1, then:
p£ss) 00 = t nr-rf *{k-1) (JI +s) ■pk-1 ■ q+ k=1 (K — 1)'
+ t Pk"{k) (P> + s)^+1 (s). k=0 K'
2.2. Time characteristics of the system
Let W(serv) (x) and W(loss) (x) be the distribution functions of the time spent in the queue by the served and dropped packets.
2.2.1. Time characteristics for a served packet
<rv) (x) — the intermediary distribution function of the time spent by the served packet in the queue, if there are i other packets in the queue before the considered one and there are j others after it. Then
w (se'v)M =(t wir'M)--^ ,
where steady-state probabilities ni (i > 0f) are defined in [37], [38]. For densities u>(sjrv)(x) = (Wi(sJerv) (x)) , we will consider several cases.
a) Consider the case when i + j + 1 > Q1, 0 < i < Q1
wtTHx) = z-ilX P+vJ Mx)+
X J k
+ [ V^^Te-™Pmin(k,J+i+1-Qi)dA(y)w{*-l+1 (x-y),
^ 0 k-
0 k=0
pmin(k,j+i+1-Q1) = {P ,k^j + i+1- Qi,
pj+1+i-Qi,k>j+i-Qi.
b) Let's move on to the case when i > Q
1
X
wfjv) (x) = e-^ pj A(x) + [V e-"y Pk dA(y)wtjv)(x - y).
, J u-n ,
0 k-°
If i + j + 1 > Q1 the threshold is not crossed, then:
X j
w.grv) (x) = e-^A(x) + [V e-^V dA(y)wtr) (x - v)-
0 k-0
The Laplase-Stieltjes transforms for derived densities. If i + j+ 1 < Q1, then:
jsfv> (.) = (-l)'!^(u + .) + V a
i" (p+s) + V Mr^- <"*■> (p + s)<4j-Us),
J- k-0
to
wfjrv) (s) = [ ^isjrv) (x)e-sxdx — Laplace-Stieltjes transform.
0
If 0 < Q1, but i + j + 1 > Q1, then:
iserv) (8) = (-1Yf+1 &(j) (p + S) . pj+i+1-Qi +
ui,j (s) = j
+V M^ °(k) (^+s) ■ Pmin(k,j+i+i-Qi). jfj-Xi (.*)■
0 k-
k-0
If i > Q1, then:
^(s) = (—1y,113+1 *(f) (V + S)-P3 + t a(k)(v + s)-pk ■J^n+1(s). 3' k=o K'
2.2.2. Time characteristics for a dropped packet
W^ (x) — the intermediary distribution function of the time spent by the dropped packet in the queue, if there are i other packets in the queue before the considered one and there are j others after it. Then
For densities u>f°ss)(^) = (W^f (x)) , we also will consider several cases.
a) The first case is when i+1+j < Q1, so the selected packet can be dropped only due to the reception of new packets in the system and overcoming the threshold value
% j ^
^ (*)= J t(-lH-e-"ydA(y)w^-k+1(x — y).
b) for the second case, when i + 1 + j > Q1, (i + 1 < Q1), several subcases should be considered:
b.1)
min(i,i+1+j-Qi) k k_1
^f°ss) (x) = t f, X r .e_^x ■ pk-1 ■ q ■ A(x)+
, k=1 (k — 1)'
% j ^
+ J e_"V ■ pmin(k„t+1+'_Qi)dA(y)wil°_k+1 (x — y).
0 k=0 k' ,
b.2) If i + 1> Q1, then:
il ) ^ ^ tjk^k 1 _
wi°j (x) = t (fr—1jje_MX ■ pk_1 ■ q ■ A(x)+
x 'J k
+ J t^ e_^y -Pk dA(y)<_;+1 (x —y).
0 k=o k'
The Laplase-Stieltjes transforms for derived densities.
a) For the case when i + j + 1 < Q1 we have
(loss)(s) = £ ( 1)k^kn(k), ^ ., ,(loss)
k=0
k.
-a(k) (V + S)-^! (S).
b) For the case when i + j + 1 > Q1, i + 1 < Q1 we obtain: b.1)
min(j,i+l+3-Q1) k_x k
J^ (s) = £ (i1) ^ a(k-1) fa + s)- pk-1 - q+
k=1
(k~1)\
+ £ Pmin{k,t+J+1-Qi)a(k) fa + s) - J^^ (s).
k=0
b.2)
(loss) (*) = £
(k-ry.
k=1
a[k V)fa + s) -pk-1 - q+
+ Pk"{k) (p + s)-u/^k+1 (s).
k=0
3. The second model
The second queuing model is also GI/M/1/<x> queuing system, shown in the figure 2, with the implemented renovation mechanism, but the threshold value Q1 determines the boundary in the queue, starting from which the dropping of customers begins and also determines the safe zone from where packets cannot be dropped.
Figure 2. Queuing system model 2
If the current number of packets in the system i is less or equal to Q1 + 1 (the threshold value Q1 has not been overcome), then none of the packets will be dropped from the queue. If the current number of packets in the system i is greater then Q1 + 1, then with probability q the packet, finishing the service and leaving the system, will drop all packets from the queue (outside the safe zone), or with probability p = 1 — q the serviced packet simply leaves the system.
Let be the steady-state probability distribution of the embedded Markov chain that the packet comming into the system will find in it i other packets (x > 0) [37], [38].
Let p(loss) and p(serv) be the probability that the received packet in the system will be dropped from the queue or will be transferred to service device.
The p(serv) is the auxiliary probability that the packet will be served if it finds other i packets in the system.
to
p(serv) = t P(serv) = 1—^0+1 ■ 1-S-7.
uPt 1 Ql+1 (1—g)(1—pg)
.(loss) =1— P(serv) =1—(1—VQ+1 .—I_
\ Ql+X (1 — g)(1—vg))
„(loss) = „ m _Q_
P ^ (1 — g)(1—pg).
3.1. Time characteristics of the system 3.1.1. Time characteristics for serviced packets
W(serv) (x) is the cumulative waiting time distribution function for the
accepted into the system packet, Wj(serv) (x) is the cumulative waiting time distribution function for the accepted into the system packet, if at the moment of its arrival there were i other packets in the system. Then:
to
w(serv) (-) = ^(¿vr tw?av) (*)■*,
p i=0
wfrv) (X) = (wfrv) (x))' — probability density function.
The auxiliary functions Wi(,s.?erv)(x) and w^rv)(x) = (^(jrv)(x)) (i,j > 0) are the distribution functions and the densities of distribution functions of the time spent by the served packet in the queue, if there were i other packets in the queue before the considered one and j others after it.
a) If i = 0, then the cumulative distribution functions Wj(serv) (x) = 1,(x = 0). b) If 0 < i < Q1 — (the safe zone is not completely filled) then the received in the system packet will be in the safe zone (cannot be dropped). Then
X
w(serv) (x) = ■ A(x) + J e_^yd(y) ■ wf^ (x — y).
0
b.1) 0 < i + j < Q1, j > 0 (taking into account the packets that came after ours), the threshold value Q1 has not been overcome in the queue, that is,
the renovation mechanism has not turned on. Then
wgerv) (*) = ^e_»* ■ ~A(x) + J t (-jf e_^ydA(y) ■ w^ (x — y).
' o k=0 '
b.2) Q1 < j + 1 (j > 0) the renovation mechanism was activated, but our packet is in a safe zone. Then
, s I.J+1 TJ _ ,,Qi_i+1 TQi_i _
w(serv) (x) = №—_^_pj_(Qi_i)+1 . A(x) + » X ^ . qe_»x . A(x) +
1+(J_(Qi_i)_1) nk+Q1_tTk+Q1_t_1 _
+ t iM — (Q1 -i)~k)-% + Ql-i~ 1 y er"'-A(x)+
x
+ J e_™dA(y) ■ wg+1 (x — y)+
0
x.i_(Qi_i)_1
+ J t ^e_"y ■pkdA(y) ■ —y)+
k=1 x
+ J t ■ PZ_Ql_ZdA(y) ■ wg_t+1 (* — y),
0 k=1_(Q1_i) K'
^ (X)= ¿J ^ — (Q1 —i) — k)■ (k + Ql —i — 1)'.'
w.
%J V /
x.]_(Qi_i)_1
+1 t ^ ■ pkdA(y) ■ wtevi+1(x—y)+
0 k=1
x j k + J t ^e_'iy ■ pt_Ql_%dA(y) ■ wS_vi+1 (x—y)-
0 k=1_(Q1_i) '
c) i > Q1 + 1 — at the time of receipt of our packet, the safe zone is filled and there are packets outside the safe zone — the renovation mechanism is enabled. Then
X
wgrv) (x) = ne_^xp ■ A(x) + J e_^ydA(y) ■ wf^ (x — y),
0
wgrv)(x) = e_^x&+1 A(x) +J t e_™■PkdA(y)-w^+1(x—y).
3' 0 k=0 '
3.1.2. Time characteristics for dropped packets
Let W(loss) (x) be the cumulative distribution functions of the time spent by the packet in the queue before dropping.
1
w (loss)^ = -i) ■£^(loss)(*K.
p
(loss)
z=0
wfoss) (x) is the conditional probability that in a time less than x the packet that has found exactly i of other packets in the system will be dropped from
the queue. The auxiliary functions wf°ss) (x) and wf°ss) (x) = (wf°ss) (x))
(i,j > 0) are the distribution functions and the densities of distribution functions of the time spent by the dropped packet in the system, if there were i other packets in the queue before the considered one and j others after it.
a) 0 < i < Q1 (that is, the system was either empty, or at least there was one free space in the safe zone)
Wfss) (x) = 0.
b) Qi <i (i>Qi + 1)
w[,0ss)(x) =/ie ^Xq-A(x) + J e dA(y) ■ w-,1ss)(x-y),
0
(loss)
w(loss) (?) £ ■h(i + i-Qi - k)A(x)+
k= 1
k=0
^ j ^ j_yU
+ J£ ■ £ ^ (i)dA(y) ■ wffk_l+1 (x).
4. The third model
Consider the GI/M/l/<x> queuing system, shown in the figure 3.
A-' O-'
Service device
Qi Ch
Figure 3. Queuing system model 3
Queue LIFO
In this section, a single-server queueing system with an infinite queue capacity and two threshold values is considered. Threshold values:
— Q1 — the threshold value in the queue, when overcoming which by the queue length packets (from Q1 + 1) will be dropped from the queue with a probability q.
— Q2 — the threshold value in the queue to which packets are dropped (i.e. packets standing in the queue up to the Q2 threshold are not dropped).
4.1. The service probability and loss probability of the received
packet
Let's introduce the probability p(serv) that the packet, entering the system, will be served, auxiliary probabilities p(serv) (i > 0) of incoming packet to be served if there were other i (i > 0) packets in the system, and auxiliary probabilities p(serv) (x) that during the time x the packet, which found exactly i other packets in the system at the moment of arrival and behind which there are j more packets, will be served
to
p(serv) = £p(serv)^,
i=0
where ^ — the stationary probabilities [37], [38].
Let's consider several cases
a) The first one, when the system is empty: p0s6rv) = 1.
b) The second case is when 1 < Q2, so p(serv) = 1.
c) The third case Q2 < i < Q1 includes two subcases:
c.1) the first subcase, Q2 + 1^i + 1+ j^Q1 + 1 — the Q1 threshold in the queue has not been overcome (taking into account the packets after the considered one), that is, the renovation mechanism has not turned on
X j
pfv) (x) = A(x) - + J£ ^e-™dA(y) - (x - y).
c.2) the second subcase, i + 1 + j > Q1 + 1 — the Q1 threshold in the queue has been overcome, so the renovation mechanism has been activated
pferv)(x) = A(x) - - pi+3+1-(Qi+1)+
{J + 1)-
»i+j-Qi ( )k
+ J £ ^^^dAty)-^^ (x-y)+
o k=0 '
% j ^
+ J £ {ji!- e-"yp*j-Qi dA(y) - pi-h (x - y)-
0 k=i+j-Q ± + 1 K-
d) the fourth case is when the Q1 threshold in the queue has been overcome at the moment of the arrival of the considered packet, (i > Qx) so the renovation mechanism has been already activated
X j
p(fr}= e-"xpJ+1 + ilb e-"ypkdA(y)-p(i^(x-y),
U +1)- 0 k=0 K. '
TO
(serv) / (serv) / \ ,
Pi p\,0 (x)ax.
0
Loss probability of the received packet
to
p(loss) = £p(loss) ^,
i=0
where p(loss) — the probability that the incoming packet will be dropped if at the moment of its arrival there were i, i > 0 other packets in the system,
and p(!oss) (x) is the probability that in time less than x the packet, before which there are i other packets in the queue and after which there are other j packets, will be dropped, i,j > 0.
a) p1loss) =0, 1 = 0,02;
b) Q2 < i < Q1 the threshold value of Q1 has not been reached at the time of receipt;
b.1) i + l+j<Q1 + 1 — (the threshold has not been crossed even taking
into account the application that came later)
y
piT) (?) = Jl e-liy dA-y) ■ ^+1 (x - y)-
0 k=0 .
b.2) i+l+j > Q1 + 1 — (the Q1 threshold was overcome due to applications after the incoming one)
г+3+1-(Ql + 1) ( )k
pg^ (x) = A(x) b Id6-1** pk-1q+
k=1 .
+ j b (~ire-"ypkdA(v)pio-k+1 (x-y)+
0 k=0 . x j k
+ i b ^ e-^vPi+j-Qi dA(y)^k+1 (x-y).
J k=i+j-Q 1 + 1 K.
c) i > Q1
, , J+1 (..„\k
(loss)
— (nr)k
1 (x) = A(x) Y Pk-1Q+
+ IY (tjf 'e-"ypk dA(y)pio-k+i (x-v);
0 k=0
(loss) (loss)
p\ = I P\ o (x)dx.
4.2. Time characteristics of the system
Let W(loss) (x) and W(serv) (x) be the cumulative distribution functions of the time spent in the system by the packet before being dropped or served.
The auxiliary functions W(serv)(x) and w^rv)(x) = (W(SjlV(x)) , wf°ss)(x)
and wf°ss)(x) = (w\serv)(x)) (i,j > 0) are the distribution functions and the
densities of distribution functions of the time spent by the served (lossed) packet in the queue, if there were i other packets in the queue before the considered one and j others after it. Then
TO
w (se'v) M = Y<T} (*)■*«
P i=0
-1 ^-VJ
p
i=0
a) If a packet enters the empty system (i = 0), it immediately starts to be served.
(serv)
w0,o ) (x) =
^ (s) = j e sxw0s,eorv) (x)d(x) = 1
0
wOy (X) = 0.
b) If the total number of packets in the system has not overcome the threshold Q2 (0 < i < Q^,, i + j + 1 < Q^^), then the considered packet will be in the safe area and the renovation mechanism is not enabled.
X
wf0rv) (x) = A(x) - + j e-^ydA(y) - wffv) (x — y).
wgrv)(x) = + jib ^dA(y) . wf^+1 (x - y),
0
4f(s) = (s + ») + ± tjf x aW (s + ri.JC-i+1 (s),
w(|°ss) (x)=0.
c) The case, when at the moment of arrival of the considered packet there were 0 < i < Q2 other packets in the system (our packet was in the safe area), but currently the total number of packets in the system is equal to i + j + 1 > Q1 (so the renovation mechanism is enabled)
wgrv) (x) = tb+^L pi+j+1-Qi A(x)+
x
(,„,)k (n(^-„))Q2-i-1 . .
e-lMx-y) +
+A(*)j b m.
0 k=1 X4^j+1-Qi ( )k
i uTe-"x pk-1 qdA(y)wt^-1+1 (* - y)+
I X-■>, \u>u
k=0
0
3 (,.„.\k
+ I i cry^1-^ dA(y)wfj—k (x),
0 k=t+j+1-Q1-1 (loss)
k.
w(;oss) (x) = 0.
d) The case, when at the moment of arrival of the considered packet there were Q2 < i < Q1 other packets in the system (our packet was out of the safe area ), includes several subcases.
d.1) The first subcase — currently the total number of packets in the system is Q2 < i + j + 1 < Q1 (the renovation mechanism is not enabled)
X
wgrv)(x) = + jib x ^dA(y) • wS-1+1 (X - y),
0
* t+j+1-Q2
wi!7)(X)=I b ^ydAy. wfo-i+1 (x-y).
* 0 kl
0 k=0
d.2) The second subcase, when currently the total number of packets in the system has overcome the threshold Q1 (i + j+1 > Q1), so the renovation mechanism is activated
wgrv) (x) = A(x)^—^— e-^y - pi+J+1-Qi + 3-
p+j+1-Q1 ( )k
+ I £ ^e-^pkdA(y) - wg-t+1 (x — y)+
0 k=o ' % j ^
+ j £ - e-^dA(y) - wfevl+1 (x — y),
0 k=i+j+1-Q1 + 1 K-
i+J+1-Qi k k-1 wf°ss) (x) = A(x) £ J7^r.pk-1 + ' k=1 (k — 1)-
*i+j+1-Q1 ( s k
+ j £ (~jrpke-"ydA(y) - w£s-l+1 (*—y)+
o k=0 ' x j k
+ j £ (JJT- - Pe-»ydA(y) - wfo-l+1 (x — y).
0 k=i+j+1-Q1 + 1 K-
e) The last case, when the threshold Q1 was overcome (i > Q1) at the moment of our packet arrival
wgrv)(x) = A(x)^—^e-^xpj+1 + j £ e-^ypkdA(y)- wg-^(x — y),
0 k=0 '
wf^ (x) = A(x) £ e-^xpk-1q+
k=1 (K—1).
x 'j k
+ j£ (jir e-™pk dA(y) - w£!+1 (*—y)-
5. GPSS simulation results
Below (see table 1) is presented a table with GPSS simulation results that was performed with the following initial parameters: threshold value Q1 = 30, arrival rate — 14 task per 1 unit of time, service rate — 16 task per 1 unit of time, and the simulation time is 100000 units of time) for different drop probabilities.
The table 2 shows the results of GPSS simulation that was performed with the following initial parameters: arrival rate — 14 task per 1 unit of time, service rate — 16 task per 1 unit of time, q = 0.01, and the simulation time
is 100000 units of time) for different threshold values. For the third model the threshold value Q2 = 10.
Table 1
Simulation results for different drop probabilities
q propability 0.0025 0.005 0.01 0.025 0.05 0.1 0.15
Generated tasks sys.1 1401525 1401566 1401134 1400127 1400915 1399127 1398795
sys.2 1400992 1401374 1401547 1400816 1401421 1400971 1401135
sys.3 1401647 1401379 1400564 1400333 1400889 1400251 1399581
Serviced tasks sys.1 1400084 1398863 1396791 1394210 1393457 1389597 1389540
sys.2 1400752 1400843 1400879 1399692 1399428 1399166 1399030
sys.3 1400537 1399411 1397201 1395975 1395643 1393555 1393104
Serviced tasks without calling the renv. mech. sys.1 1379233 1381969 1385859 1388162 1388647 1386899 1387651
sys.2 1378347 1381669 1385318 1388493 1387780 1391338 1391897
sys.3 1379887 1382616 1385828 1389605 1390628 1390814 1391166
Dropped tasks sys.1 1436 2698 4332 5917 7456 9530 9249
sys.2 240 527 663 1117 1984 1803 2104
sys.3 1091 1967 3357 4357 5240 6696 6472
Service Probability sys.1 0.9990 0.9981 0.9969 0.9958 0.9947 0.9932 0.9934
sys.2 0.9998 0.9996 0.9995 0.9992 0.9986 0.9987 0.9985
sys.3 0.9992 0.9986 0.9976 0.9969 0.9963 0.9952 0.9954
Drop Probability sys.1 0.0010 0.0019 0.0031 0.0042 0.0053 0.0068 0.0066
sys.2 0.0002 0.0004 0.0005 0.0008 0.0014 0.0013 0.0015
sys.3 0.0008 0.0014 0.0024 0.0031 0.0037 0.0048 0.0046
Average queue length sys.1 6.0930 5.9230 5.7090 5.5240 5.4820 5.3080 5.2360
sys.2 6.1800 6.0780 6.0220 5.8580 5.9530 5.7980 5.8550
sys.3 6.1230 5.9360 5.7330 5.5720 5.5560 5.4120 5.3290
Maximum queue length sys.1 92 71 63 67 54 46 43
sys.2 92 64 61 65 60 51 49
sys.3 92 71 71 67 54 46 43
Average waiting time sys.1 0.497 0.483 0.467 0.453 0.449 0.437 0.431
sys.2 0.503 0.495 0.491 0.478 0.485 0.473 0.478
sys.3 0.499 0.484 0.469 0.456 0.454 0.444 0.438
Table 2
Simulation results for different threshold values
Threshold 10 20 25 30 40 50 75
value Q-y
Generated tasks sys.1 1399202 1401573 1401188 1401134 1399645 1400335 1400451
sys.2 1399603 1400523 1399393 1401547 1402003 1400032 1399596
sys.3 1399603 1400753 1400647 1400564 1399680 1400321 1400448
Serviced tasks sys.1 1368353 1389618 1393927 1396791 1398462 1399917 1400367
sys.2 1387180 1397457 1397721 1400879 1401813 1399986 1399562
sys.3 1387180 1393344 1395743 1397201 1398764 1399969 1400374
Serviced tasks sys.1 1166280 1343186 1370099 1385859 1394747 1398969 1400319
without calling sys.2 1145456 1336931 1365038 1385318 1396545 1398819 1399341
the renv. mech. sys.3 1145456 1346681 1372422 1385828 1395050 1399021 1400326
Dropped tasks sys.1 30833 11955 7261 4332 1176 407 83
sys.2 12423 3065 1672 663 190 42 33
sys.3 12423 7409 4902 3357 916 337 73
Service Probability sys.1 0.9780 0.9915 0.9948 0.9969 0.9992 0.9997 0.9999
sys.2 0.9911 0.9978 0.9988 0.9995 0.9999 1.0000 1.0000
sys.3 0.9911 0.9947 0.9965 0.9976 0.9993 0.9997 0.9999
Drop Probability sys.1 0.0220 0.0085 0.0052 0.0031 0.0008 0.0003 0.0001
sys.2 0.0089 0.0022 0.0012 0.0005 0.0001 0.0000 0.0000
sys.3 0.0089 0.0053 0.0035 0.0024 0.0007 0.0002 0.0001
Average queue length sys.1 4.564 5.273 5.5330 5.7090 5.9110 5.934 6.158
sys.2 5.069 5.7 5.8540 6.0220 6.0780 6.014 6.089
sys.3 5.069 5.37 5.5630 5.7330 5.9210 5.933 6.158
Maximum queue length sys.1 67 64 71 63 80 76 89
sys.2 67 75 62 61 64 76 102
sys.3 67 75 59 71 80 76 89
Average waiting time sys.1 0.381 0.433 0.454 0.467 0.484 0.485 0.502
sys.2 0.418 0.466 0.479 0.491 0.496 0.491 0.497
sys.3 0.418 0.441 0.456 0.469 0.485 0.485 0.502
6. Conclusion
Based on the simulation results 1, the following conclusions can be drawn. The largest number of dropped packets, as expected, is observed in the first model, the smallest — in the second model (due to the safe zone). The third model shows an average result compared to the first and the second models. The largest number of serviced packets is in the second model, then — in the third model. The smallest number of serviced packets is in the first model.
The probability of a packet to be dropped is about five times greater for the first model than for the second model, and 20-30 percent more than for the third model.
The average waiting time for the second model is about 5-10 percent greater than the same characteristic for the first and third models.
As the value of the renovation probability q increases, the drop probability increases for all three models, and the service probability decreases accordingly. Also, with an increase of the renovation probability q, both the average and maximum queue lengths decrease, and the average waiting time also decreases.
Based on the simulation results 2, the following conclusions can be drawn. With an increase of the threshold value Qx responsible for switching on the renovation mechanism, the number of dropped packets decreases for all three models (the second model is characterized by the smallest number of dropped packets), the service probability increases to unity (the second model), and the drop probability decreases almost to zero. The average and maximum queue lengths increase, and the values for the first and third models become approximately the same. The average waiting time also increases, and again for the first and third models, the values become approximately the same.
The third model, which generalizes the first and the second models, shows average results compared to the above models, and is more preferable for use as a queue length management model.
Acknowledgments
The publication was funded by RFBR according to the research projects No. 20-07-00804.
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For citation:
I. S. Zaryadov, H. C. C. Viana, T. A. Milovanova, Analysis of queuing systems with threshold renovation mechanism and inverse service discipline, Discrete and Continuous Models and Applied Computational Science 30 (2) (2022) 160-182. DOI: 10.22363/2658-4670-2022-30-2-160-182.
Information about the authors:
Zaryadov, Ivan S. — Candidate of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics of Peoples' Friendship University of Russia (RUDN University); Senior Researcher of Institute of Informatics Problems of Federal Research Center "Computer Science and Control" Russian Academy of Sciences (e-mail: zaryadov-is@rudn.ru,
phone: +7(495)9550927, ORCID: https://orcid.org/0000-0002-7909-6396, ResearcherlD: B-8154-2018, Scopus Author ID: 35294470000) Viana, Hilquias C. C. — PHD student of Department of Applied Probability and Informatics of Peoples' Friendship University of Russia (RUDN University) (e-mail: hilvianamat1@gmail.com, phone: +7(495)9550927, Scopus Author ID: 57212930802)
Milovanova, Tatiana A. — Candidate of Physical and Mathematical Sciences, Lecturer of Department of Applied Probability and Informatics of Peoples' Friendship University of Russia (RUDN University) (e-mail: milovanova-ta@rudn.ru, phone: +7(495)9550927, ORCID: https://orcid.org/0000-0002-9388-9499, Scopus Author ID: 26641495400)
УДК 519.872:519.217
PACS 07.05.Tp, 02.60.Pn, 02.70.Bf
DOI: 10.22363/2658-4670-2022-30-2-160-182
Анализ систем массового обслуживания с пороговым
/•* о о
механизмом обновления и инверсионном дисциплинои
обслуживания
И. С. Зарядов1,2, Илкиаш К. К. Виана1, Т. А. Милованова1
1 Российский университет дружбы народов, ул. Миклухо-Маклая, д. 6, Москва, 117198, Россия 2 Институт проблем информатики, Федеральный исследовательский центр «Информатика и управление» РАН, ул. Вавилова, д. 44, кор. 2, Москва, 119333, Россия
Аннотация. В работе представлено исследование трёх систем массового обслуживания с пороговым механизмом обновления и инверсионной дисциплиной обслуживания. В модели первого типа пороговое значение отвечает только за активацию механизма обновления — механизма вероятностного сброса заявок. Во второй модели пороговое значение не только включает механизм обновления, но и определяет в накопителе границы области, из которой поступившие в систему заявки не могут быть сброшены. В модели третьего типа, обобщающей предыдущие две модели, используются два пороговых значения: одно для активации механизма сброса заявок, второе — для задания безопасной зоны в накопителе. На основе полученных ранее результатов представлены основные вероятностно-временные характеристики рассмотренных моделей. С помощью имитационного моделирования проведён анализ и сравнение поведения изученных моделей.
Ключевые слова: система массового обслуживания, активное управление очередью, механизм обновления, пороговое значение, временные характеристики, GPSS
