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ALGORITHMIC CALCULATION OF STATIONARY DISTRIBUTION FOR MULTIPHASE QUEUE WITH COMMON FOR PHASES
SERVERS IN DISCRETE TIME
Xatiana V. Rykova,
Fraunhofer Heinrich Hertz Institute, Berlin, Germany, tatiana.efimushkina@hhi.fraunhofer.de
Keywords: queuing system, tandem queue, balance equations, performance measures.
Among the advantages of the Next Generation Mobile Networks (NGMN), including fourth-generation (4G) mobile technologies, such as WiMAX, WiMAX 2, LTE, LTE-Advanced and developing currently fifth-generation (5G) networks are greatly enhanced user-experience, low latencies in both processing and signal transmission, extension of the provided network services and interoperability and service continuity with the existing networks of previous generations. In this framework resource allocation in NGMN networks becomes an important and crucial problem that directly influences the potential capacity and coverage improvements given the constant growth of traffic volumes and change of its structure especially for data transmission in downlink channel, i.e. from the base station to the user equipment.
Heterogeneous networks are characterized by placement of base stations that transmit at high power levels overlaid with relay nodes with substantially lower power levels. Heterogeneous networks are considered to be a promising solution due to improvement of system capacity and coverage, support of high data rates and reduction of deployment costs.
In this paper a tandem queue is proposed and investigated with servers distributed among the phases. The given queuing system is considered in discrete time with buffers at the phases of limited and unlimited capacity. The particular use case of the tandem queue is a model in heterogeneous LTE network with two types of nodes: base stations and relay nodes for performing analysis of various resource allocation schemes. The given queuing system addresses the practical computational problems of NGMN heterogeneous networks, including the ones with cross-layer adaptation, which are characterized by distribution of overall number of cell resources between the base station and relay nodes in every time slot.
Information about author:
Rykova Тatiana Vladimirovna, Research Associate, Fraunhofer Heinrich Hertz Institute, Berlin, Germany
Для цитирования:
Рыкова Т.В. Алгоритмический расчет стационарного распределения многофазной СМО с общими для фаз приборами в дискретном времени // T-Comm: Телекоммуникации и транспорт. 2017. Том 11. №12. С. 71-76.
For citation:
Rykova T.V. (2017). Algorithmic calculation of stationary distribution for multiphase queue with common for phases servers in discrete time. T-Comm, vol. 11, no.12, рр. 71-76.
In [1 ] a new multiphase queuing system in discrete time with servers distributed between phases is proposed and described
(Geom(1(q}Bin{q}c:Mh <«|S/»(q)il/SAT,, / = 1,2, with a
set of structural parameters ¡A(q), b(q), c, ge, (q), r, d{q)],
describing the functioning of the system, and vector of servers c" at the phases, that is defined by the distribution £c(q) taking
into account c prea I located to the phases servers and an overall number of C servers. Further in the paper we will use notations defined in [1].
The given queuing system addresses the practical computational problems of NGMN heterogeneous networks, including the ones with cross-layer adaptation, which are characterized by distribution of overall number of cell resources between the base station and relay nodes in ever}' time slot [2-4].
The particular case of tandem queue was investigated in [5, 6], whereas the tandem queue with one server was described in [7].
Model description
Let M be a number of phases in a tandem queue, I < M < vo. We consider a tandem queueing model, in which the capacity of the buffer at the phase m is r , 0 < rm < oo. Further
in model description if not mentioned explicitly m = I, M and m corresponds to the number of the phase. The homogeneous requests arrive at the buffer of the phase m ■
After being serviced at the phase m a request may either leave the landem queue with a given probability or continue the serving process by arriving at the buffer of the next consequent
phase /w+l, m = \,M — 1; in case when m-M all the serviced requests leave the tandem queue.
The structure of the queuing model is depicted in Fig.l. Here and further, a dot in place of an index means a full sum of the variable by this index.
We consider that from the overall number of servers c,
c< oo in a tandem queue, Cm servers are constantly preallocated to the phase m, cr := (cl,c2,.,.,cM), while the rest servers c-c. are distributed randomly according to the procedure described further in the paper; 0 < c,„ < c, 0 < < c > taking
into account that a if c ~c.
ffl *
The case cm =0 means no constant preal location of the
servers to the phase m; if c, = 0, then a whole set of servers is
distributed among the phases according to a given procedure; in case when c.~c we obtain a common tandem queue with a
fixed number of servers cm at the phase m without any further
redistributions. Generally, among the random server's distribution between the phases there is also a deterministic one, e.g. the servers are allocated in proportion to the number of requests at the buffers.
The system functions in discrete time in time slots of constant length h, h>0 as illustrated in Fig,2. We assume that all the changes in the system occur at time moments n/u n = 0,1,... and n is a half-closed interval \nh, (n+1)/?}.
Fig, I. A tandem queue in discrete time with an overall set of servers distributed among the phases
We consider the following order of possible events in a iandem queue at time slot n\
- distribution of the servers c — c, among the phases;
- requests are being serviced at the phases, starting from phase M to phase 1; consequently releasing the spaces in the buffers in ease of successful transmissions and leaving the tandem queue or arriving at the free buffer spaces of the next phases; (it phase M is reached, the successfully serviced request leaves the system);
- arrival of new requests at the phases of the tandem queue;
- the state of the system is fixed.
The system stale at time slot n coincides with the state at time moment nh after all the active events are finished.
nh (w + l)/?
) | Lime slot >1-1 ) I lime slot n ) I
I-• ' ' I- 1 - I ' ' ' ■
0 Distribution of the servers c — c. among the phases m\-M * Requests are serviced at the phase m
Spates in the buffer of phase m are released, requests leave tandem queue (fn-M) or arrive at the phase m+t
!lfm>l, then m: - m-1 and transition to gtherwise Arrival of new requests at the phases i i
System stale is fixed
Fig, 2. Order of events in the proposed tandem queuing model in discrete time
The system state in time slot n coincides with the slate in time moment nh after termination of all the active system events.
The given requests service order, starting from phase M and sequentially descending to phase 1, is a natural proposal that minimizes request losses in case of a presence of limited buffer capacities at phases of a tandem queue. When rm= oo,
m = \,M, the service order does not have any impact on the losses of the requests.
Balance equations
The functioning of the system in its general case 111 when
'',„ < hi = I, Af ■ is described by the homogeneous process at time moments nh< with the
state space:
Q=U = (1)
Based on the given assumptions the process £ , «>0, is a
homogeneous Markov chain, which states (1) £w, w>0,
communicate and constitute to one ergodie class without subclasses. It is considered that introduction of dependence of
the structural parameters on the Markov chain state in the previous time slot does not result in emergence of new states that cannot commun icate.
In the given conditions there exist a final distribution ([q]:= I im (?{<!;„ },q e Q)> lq] > 0, qeQ, that does not depend
on initial one and coincides with a stationary probability distribution [8].
The stationary probability distribution
([q]:= [q^q2.....qM], q e Q), can be found from the balance
equations of the order |q| and rank [q| - ]:
iq'l = I[q]"(q.q'K£Q' (2)
qsQ
with a normalizing equation
Here £i(q,q') - is a transition probability of a Markov chain £ at one time slot from the state q into the slate q': a(q,q'):=<Pfe, =q%,_i =q}> q,q'eQ, «>1.
Let us introduce a set of stales of the Markov chain Q{q'), from which the state q' can be reached in one time slot:
Q(q'):= {q,qeQ|a(q,q')*o}. Q(q')cQ-q'eQ'
Definition 1. A set of realizations of three successive events: ( distribution of servers among the phases; termination of
service of the requests at phases and arrival of the successfully serviced requests to the buffers of the next phases; arrival of new requests to the phases of queuing system J, which can happen in
a queuing system during one time slot at state q of the system in previous time slot, is defined as a combination of events and denoted by <y(q).
An example of events combination is <y(q) = (cF;s,v;k). A set of all the combinations of events cy(q) will be denoted by Q(q).
Definition 2. A combination of events ®(q) is said to be congruent lo a state q' and denoted by <y(q,q'), if as the result of fc)(q) a queuing system goes in one time slot from the state q to the state q' (q *h> >q').
A set of combination of events *y{q,q') wili be denoted by Q(q,q'); and consequently Q(q,q')c Q(q), h
U(q} ■
qeO:il(q.q>0
Based on the introduced assumptions: a(q,q')= I(PMq,q')}.
whereas equations (2) are represented as follows:
[q'l- I [q] |<P{%q')}'q^Q»
Writing the balance equations explicitly could hardly be realized for the investigated queuing model due to a large number of random events, occurring in the system; however, the
balance equations (2) can be obtained algorithmically, which reduces the computation process by finding Q(q'), all congruent combinations of events <y(q,q') and, therefore, obtaining the equations (4) (coefficients «(q,q') in (2)).
Algorithmic computation of stationary probability distribution
Let X be a set that consists of elements \T = where xt ~ is an integer that takes values from an ordered
ascending set x( |r'in,..r'""} from a minimal value A'(mm ,
increasing to a maximal value , i=l,I, X=]^[X. >
i=i
1=1
Let x be an / - ordered element in X.
j
Definition 3. Let us define as initial and final elements of a set X the elements (x""n and
(jef8*,;t™x,.. respectively.
Definition 4, A set X will be called ordered, if its elements are arranged sequentially, starting from the initial to the final one in accordance with the following algorithm:
Let the sets Q, C', K be ordered. For each pair q and c'(q ) we consider an ordered set S(q,c'(q)) of all the variants of serviced requests at the phases during one time slot. An ordered set of all the variants of arriving requests v at the buffers of the next phases (for the last phase - the requests leaving the queuing system) from the number of serviced requests s at all the phases of the queuing system will be denoted by V(s):={v:O<v„,<Sm_1,m = 2j7,v1=0}.
The constructive procedure of forming a set Q(q') and an array of non-zero matrix coefficients A = ||a(q,q'))| (2) is
represented by the following algorithm.
Algorithm 2. BEGIN
for q = inilial element Q to final element Q do for q = initial element Q to final element Q do
o(q,q'):=0
for c' = initial element C' to final element O do
for S = initial element S(q,c'} to final element S(q,c ) do
for V = initial element V(s) to final element V(s) do
for k = initial element IS. to final element K do
if q-s + (r-q + s°v + k) = q' then {c';s,v,k)eQ(q,q'). q € Q(q').
«(q. q') - i')+(q) ■(<i) ■ ) ■ Vq)
else q e Q(q') end if
end for k end for V end for $ end for c' end for q end for qf END
Theorem 1. The matrix A = ||c/(ti n'll| , formed
II Vn'T 1lq,qFeQ
according to algorithm 2, uniquely defines the balance equations (2).
Proof. The left side of the relation (5) forms the state, which can be reachcd from q by servicing s requests at the phases (with probability As(q)) for c' allocated servers (with probability gc<(q)) and arrival of v requests (with probability i/s v(q)), remaining in the system, arriving from the previous phase, arrival of k new requests (with probability wk(q)), or
completely filling the buffer of the phase, if the overall number of arrived at the buffer requests exceeds the number of available places in it.
Thus, the search of all combinations of events ft>(q) allows us to find with the help of condition (5) all the elements of the set Q(q') and all congruent combinations of events of the set
C!(q,qF).
The equation (6) provides an accumulation of transition probability from q to q' for all (y(q,q')>
Consequence l.The number of checks of condition (5) is equal to |q|2|C'|s(^,c^|v(,v|k| •
Proof follows immediately from Algorithm 2
r >
Consequence2. The set Q(q')' q'eQ, obtained by Algorithm 2 is ordered.
Proof. The set Q(q') is ordered because it is formed from an ordered set Q by removing individual elements that do not
correspond to condition (5). ♦
The algorithmically obtained balance equations along with the normalizing equation (3) allow to use numerical methods, including methods of computation with sparse matrices [91 for finding stationary distribution ([q |, q e q) .
References
1 Efimushkina, T.V. (2015), Tandem queue in discrete time with distributed among phases set of servers. T-Comm, vol, 9, no. 7, pp. 60-68.
2 Efimushkina T., Gabbouj M. (2014), Cross-layer adaptation-based video downlink transmission over LTE: Survey, Coramunications in computer and information science, no. 279, Berlin, Heidelberg: Springer, pp. 101-113.
3 Efimushkina T., Gabbouj M., Samuylov K. (2014), Analytical model in discrete time for cross-layer video communication over LTE. Automatic control and computer sciences, vol. 48, no. 6, pp. 345-357.
4 Efimushkina T., Samuylov K. Analysis of the resource distribution schemes in LTE-advanced relay-enhanced networks. Communications in computer and information science, no. 279, Berlin, Heidelberg: Springer, pp. 43-57,
5 Efimushkina T. (2015), Performance evaluation of a tandem queue w ith common for phases servers. Proc. of the' 18th Internationa! scientific conference "Distributed computer and communication networks: control, computation, communications" DCCN-2015 (Octobcr 19-22, 2015, Moscow, Russia), ICS RAS, Moscow: Technosphere, pp. 44-51.
6 Efimushkina, T.V. (2015), Analysis of finite capacity queue with two phases in discrete time with distributed among phases set of servers. Abstracts of the IX International scientific-technical conference "Information society technologies". 24 March 2015, Moscow, Media Publisher, pp. 42-43.
7 Efimushkina, T.V. (2015), Analysis of multiphase queue With a single server. Abstracts of the IX International scientific-technical conference "Information society technologies", 24 March 2015, Moscow, Media Publisher, pp. 43-44.
8 Feller, W. (1968), An Introduction to Probability Tlieoiy and its Applications, vol. 1, NY: John Wiley & Sons, 3d ed.
9 Tewarson, R.P. (1977), Sparse Matrices, Academic Press, 160 p.
АЛГОРИТМИЧЕСКИЙ РАСЧЕТ СТАЦИОНАРНОГО РАСПРЕДЕЛЕНИЯ МНОГОФАЗНОЙ СМО С ОБЩИМИ ДЛЯ ФАЗ ПРИБОРАМИ В ДИСКРЕТНОМ ВРЕМЕНИ
Рыкова Татьяна Владимировна, Fraunhofer Heinrich Hertz Institute, Берлин, Германия, tatiana.efimushkina@hhi-extern.fraunhofer.de
Aннотация
Преимуществами мобильных сетей последующих поколений NGMN (Next Generation Mobile Network), включающих сети 4G (WiMax, WiMax 2, LTE, LTE-Advanced) и разрабатываемые сегодня сети пятого поколения, являются значительное повышение эффективности функционирования, снижение задержек обработки и передачи, расширение предоставляемых услуг при обеспечении интеграции с сетями предыдущих поколений. При этом задача повышения эффективности распределения ресурсов для передачи информации в сетях NGMN в целях обеспечения требуемого качества функционирования сети и предоставления услуг является крайне актуальной в условиях появления новых услуг связи и приложений мультимедиа, продолжающегося роста объемов трафика и изменения его структуры в особенности для нисходящего канала передачи данных от базовой станции к оборудованию пользователя. Гетерогенная сеть характеризуется размещением традиционных базовых станций наряду с использованием ретрансляционных станций с более низкой мощностью передатчика. Гетерогенные сети применимы для поддержания высоких скоростей передачи данных, а также для предотвращения затухания сигнала на границе соты. Исследуется предложенная ранее автором многофазная система массового обслуживания с распределяемым между фазами множеством общих приборов. В общем случае система массового обслуживания рассматривается в дискретном времени с буферными накопителями конечной и бесконечной емкости на фазах многофазной системы. Данная система массового обслуживания может использоваться, в частности, в качестве модели взаимодействия базовой и ретрансляционных станций в соте гетерогенной сети LTE в целях анализа различных схем распределения ресурсных блоков между ними. Введение данной системы массового обслуживания вызвано практическими задачами расчета гетерогенных сетей NGMN, в том числе, с межуровневой адаптацией, в которых общие ресурсы передачи распределяются на каждом такте между базовой станцией и ретрансляционными станциями.
Ключевые слова: система массового обслуживания, многофазная система, система уравнений равновесия, функционирование системы. Литература
1. Ефимушкина Т.В. Многофазная СМО в дискретном времени с распределяемым между фазами множеством приборов // T-Comm: Телекоммуникации и транспорт. 2015. Том 9. №7. С. 60-68.
2. Efimushkina T., Gabbouj M. Cross-Layer Adaptation-Based Video Downlink Transmission over LTE: Survey // Communications in Computer and Information Science. No. 279. Berlin, Heidelberg: Springer, 2014. Pp. 101-113.
3. Efimushkina T., Gabbouj M., Samuylov K. Analytical model in discrete time for cross-layer video communication over LTE // Automatic Control and Computer Sciences. 2014. V. 48. No. 6. Pp. 345-357.
4. Efimushkina T., Samuylov K. Analysis of the Resource Distribution Schemes in LTE-Advanced Relay-Enhanced Networks // Communications in Computer and Information Science. No. 279. Berlin, Heidelberg: Springer, 2014. Pp. 43-57.
5. Efimushkina T. Performance evaluation of a tandem queue with common for phases servers / Proc. of the 18th International Scientific Conference "Distributed Computer and Communication Networks: Control, Computation, Communications" DCCN-2015 (October 19-22, 2015, Moscow, Russia), ICS RAS, 2015. М.: Техносфера, 2015. Pp. 44-51.
6. Ефимушкина Т.В. Исследование двухфазной системы конечной емкости в дискретном времени с распределяемым между фазами множеством приборов // Тезисы докладов IX Международной отраслевой научно-технической конференции "Технологии информационного общества", 24 марта 2015 г. М.: Медиа Паблишер. С. 42-43.
7. Ефимушкина Т.В. Анализ многофазной СМО с единственным прибором // Тезисы докладов IX Международной отраслевой научно-технической конференции "Технологии информационного общества", 24 марта 2015 г. М.: Медиа Паблишер. С. 43-44.
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9. Тьюарсон Р. Разреженные матрицы. М.: Мир, 1977. 188 с.
Информация об авторе:
Рыкова Татьяна Владимировна, научный сотрудник Fraunhofer Heinrich Hertz Institute, Берлин, Германия
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