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LIMIT THEOREM FOR CLOSED QUEUING NETWORKS WITH EXCESS OF SERVERS
G.Tsitsiashvili
IAM FEB RAS, Vladivostok, Russia guram@iam.dvo.ru
ABSTRACT
In this paper limit theorems for closed queuing networks with excess of servers are formulated and proved. First theorem is a variant of the central limit theorem and is proved using classical results of V.I. Romanovskiy for discrete Markov chains. Second theorem considers a convergence to chi square distribution. These theorems are mainly based on an assumption of servers excess in queuing nodes.
1. INTRODUCTION
In [1], [2] problems of a formulation and a proof of the central limit theorem for queuing systems and networks are considered. At the international conference "Probability theory and its applications$" the author of this manuscript had useful and productive discussion with A.A. Nazarov stimulated an interest to this problem.
In this paper a model of closed queuing network with an excess of servers in its nodes and with singular service time or geometrically distributed service time is considered. For loads of this model nodes the central limit theorem is formulated and is proved using classical results of V.I. Romanovskiy [4] for discrete Markov chains. It is suggested to define parameters of limit normal distribution using Monte-Carlo simulations of single customer motion along nodes of the network. This approach allows to decrease calculation time significantly because it is not necessary to simulate process of loads in nodes of this network.
The central limit theorem is complemented by a fact of a convergence to chi-square distribution for large number of customers.
2. FORMULATION AND PROOF OF MAIN RESULTS
In this paper we consider closed queuing network S with the set of nodes N = {1,..., n} and with m customers. Assume that in each node there are m servers. The network works in discrete
II wn
time 0,1,..., and service time of each customer equals unit. Suppose that 0 = IpijH.^ is route matrix of the network S which satisfies the condition
(A) for any i,j EN there are i1,..,i kEN so that the product di:ii • diii2 • ... •Qik_1,ik^
K.j>0 ' ' "
Consider discrete Markov chain xt, t = 0,1,... with the set of states Wand with the matrix of transition probabilities0 . From Condition (A) we have that this chain is ergodic [3, chapter XV], denote its limit distribution by nt, i E N . This distribution does not depend on initial state x0 of the chain xt, t = 0,1,... Designate Tt(T) =# {t: xt = i, t = 0,..., T} sojourn time of the chain xt in the state i on time interval t = 0,... ,T. Introduce n - dimensional random
vector ,t EN . In [4, chapters IV, V] it is proved that there is n - dimensional and
normally distributed random vector R = (r£), i E N with zero mean and with covariance matrix B so that for any real numbers t1, ...,rn independently on initial state Xq for T —> ^ we have the convergence
T. , iEN)^P(ri<ri, iEN). (1)
Return now to closed queuing network S and enumerate customers of the network by 1, ...,m. Denote xJt , t > 0 ,j = 1, ...,m trajectories of the network S customers along its nodes. These trajectories are independent Markov chains with the set of states N and with matrix of transition probabilities 0. Assume that rJi (T) =# { t ■ xJt = i, t = 0,... ,T] is sojourn time of the
customer j in the node i on time interval 0,...,T. Introduce random vectors ' ' ,i EN,
which are independent because trajectories of different customers are independent also. Analogously with Formula (1) obtain for arbitrary real T1,...,xm for T —>
p^rypL<T. , iEN^Ptt < r£ , iEN) , (2)
where n - dimensional random vectors r? ,i E N, are independent and have normal distribution with zero mean and covariance matrix B . Consequently for T —> we have
p(l™1T-^pL<ri, iEN^P(Ri<ri, iEN) , (3)
where (Rt, iEN) is n - dimensional and normally distributed random vector with zero mean and with covariance matrix mB . Formula (3) may be rewritten for T —> as follows
p^m-rnntT <Ti, iEN)^P(ri<^Ti, iEN) , (4)
Here random variables Ti(T) = E7jl=i^Ji(T) , iEN designate total loads on iEN of the network S on time interval t = 0, ...,T . It is clear that Ti(T) = Et=omt(^), where mi(t) is a number of servers of the node i, busy by customers at moment t.
So the central limit theorem for discrete Markov chain with finite set of states may be transferred onto random vector (Ti(T), i E N) consisted of loads of the network S nodes. This result may be generalized in different directions. Assume that random service time ^ in the
node i has geometrical distribution P(^i = k) = (1 — ai)al[~1,k = 1,2,..... Then redefining
route matrix 0 by the formulas:
^=17? ■ J*1 ■ JEN , (5)
it is possible to obtain results represented by Formula (4). Moreover Formula (4) may be transformed into formula which characterizes loads in some but not all nodes 1,...,n1 < n of the network S for T —* <x> :
P (Tl(T)™lT < T. , 1<i<n1)^P(ri<^Ti, 1<i<ni) , (6)
To estimate covariance matrix Busing Formulas (1) - (4), (6) it is possible to estimate the matrix \\cov{Ti(T),Ti(T))\\ii_i by Monte-Carlo simulations with sufficiently large T . This
estimate is based on independent realizations of Markov chain xt, t = 0,1, ...,T . Formulas (4), (6) are constructed for total loads of the network S nodes and are not connected with a motion of single customer.
Consider now an aggregation of nodes in this model. An aggregation of nodes in closed queuing network leads to very complicated procedure because of difficult symbolic calculations. So it is interesting to consider this problem from a view of the central limit theorem. For a simplicity of a consideration divide the set of nodes N = {1,...,n} into two subsets = {1,...,n1}, N2 = {n1 + 1, ...,n}, 1 <n1 <n . Then total loads T1(T) , T2(T) on the sets N1, N2 of nodes are defined by the equalities
T1(T) = £tENl Tt(T), T2(T) = £tENl Tt(T). Consequently covariance matrix cou^rfc(r),Tr(r)^| may be calculated by
covariance matrix \\cov(Ti(T),Ti(T)')\\ using simple equalities
cov{Tk(T),Tr(T)) = ^ cov(Ti(T),Tk(T)) , k,r = 1,2.
iENk,kENr
And in a case of two nodes we have
cov(T1(T),T1(T)) = cov(T2(T),T2(T)) = -cov(T1(T),T2(T)) = DT1(T) . Consequently an aggregation of nodes in closed network leads to simple and clear formulas for covariances of loads in aggregated nodes. And the central limit theorem for nodes of the network S are transformed into the central limit theorem for aggregated nodes of this network: for any real r1 , r2 and T —>
p (Tk(T)~mJ;iEN*ni <T« ,k = 1,2) - p&^n ,k = 1,2) .
Assume that service times in customers of all nodes are unit and nt ,i E N is the distribution of ergodic and stationary Markov chain xt, t = 0,1,... Introduce random variables
bm(0) = ^ m
i=1
Then from [5, chapter III, pp. 169-172] we obtain the following statement. For any positive t and for m — <x>
P(bm(0) <t) — P(xLi < t).
Here mi(0) is the number of customers at the node i at the moment 0 and Xn-i is random variable with chi square distribution and with n-1 degree of freedom. This statement may be generalized onto arbitrary stationary stochastic sequence xt, t = 0,1,..., with stationary distribution nt ,i E N .
The author thanks A.A. Nazarov for useful discussions.
REFERENCES
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2. Moiseeva S.P., Nazarov A.A. Method of sifted flow for investigation of non Markovian queuing systems with unbounded number of servers. Abstracts of International conference "Probability theory and its applications$",\;$ in commemoration of the centennial of B.V. Gnedenko. M: LENAND. 2012. P. 200 - 201. (In Russian).
3. Feller W. Introduction to prabability theory and its applications. T. 1. I.: Mir. 1984. (In Russian).
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