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Tsitsiashvili G., Osipova M. RT&A, No 1 (48) GENERALIZATION AND EXTENSION OF BURKE THEOREM_Volume 13, March 2018
Generalization and Extension of Burke Theorem
Gurami Tsitsiashvili, Marina Osipova
Far Eastern Federal University Institute of Applied Mathematics FEB of RAS mailto:[email protected], [email protected]@iam.dvo.ru, [email protected]
Abstract
A simplification of Burke theorem proof [1] and its generalizations for queuing systems and networks are considered. The proof simplification is based on the fact that points in output flow take place in moments when Markov process of customers number in queuing system has jumps down. In such way it is possible to obtain a property of the mutual independence of the flow into disjoint periods of time and to calculate intensity of output flow.
Keywords: an output Poisson flow, the Jackson network, the Burke theorem
1 New proof of Burke theorem
In [1] we prove the following statement: in queung system M|M|n|ro in stationary state, the output flow has the same distribution as the input flow. Recently, however, interest in the study of flows in queuing systems is increased. Now it is necessary to give a more compact and convenient for generalizations proof of this theorem.
A random sequence of points will be called a Poisson flow with continuously differentiable intensity A(t), t > 0, if the following conditions are satisfied [2, page 12, 13], [3, page 20, 35 -- 37]:
a) the probability of the existence of the point of flow on the time interval [t, t + h) does not depend on the location of the points of the stream up to the time t (this property is called lack of follow-through and expresses the mutual independence of the flow stream into disjoint periods of time);
b) the probability that a flow point appears in the semi-interval [t,t + h) is A(t)h + o(h),
h ^ 0;
c) the probability of occurrence of two or more flow points in the range [t, t + h) is o(h),
h^0.
Let the system An = M|M|n|^ of the Poisson input flow has an intensity A > 0, and the service time has an exponential distribution with the parameter ^ > 0, 1 < n < ro. Denote Pk,n(t), k > 0, distribution of the number of customers in the system at the time t.
Theorem 1. The output flow in queuing system An is Poisson with intensity a(t) = H0<fc nPk,n(t)min(k,n). Let the output flow Tn = {0 < t1 < t2 < ••• } be An described by a random function yn(t) equal to the number of points of this stream on the segment ([0, t). Denote xn(t) the number of customers in the system An at the time t. It is known that a random process xn(t) is Markov process (of death and birth of [3, Chapter II, $$]), with each point of the Tn flow corresponding to the time of the jump down process xn(t). Therefore, the output flow Tn satisfies the condition a). In turn, the condition b) follows from the equalities:
Tsitsiashvili G., Osipova M.
GENERALIZATION AND EXTENSION OF BURKE THEOREM
RT&A, No 1 (48) Volume 13, March 2018
P(yn(t + h)= yn(t) + 1) = YJnk=1 P(yn(t + h) = yn(t) + 1/xn(t) = k)Pkn(t) +
+P(yn(t + h= yn(t) + 1/xn(t) > n) £k>n PKn(t) =
= £l=i Pk,n(t)(kVh + ok(h)) + £k>n Pkn(t)(n^h + o0(h)) = a(t)[ih + o(h), where for h ^ 0 we have ok(h)/h ^ 0, k = 0, ...,n, o0(h)/h ^ 0,
o(h) = rk=i Pk,n(t)ok(h) + £k>n Pk,n(t)°o(h), Oo(h)/h ^ 0. Thus, the output flow Tn satisfies the condition b). Check of condition c) is quite obvious.
Theorem 2. In queuing system An, when the ergodicity condition X < n is satisfied and the process xn(t) is stationary, the output flow is Poisson with intensity X. Due to the steady state of the process xn(t) the distribution of the number of customers Pk>n(t) = Pkn in the queuing system An at p = X/^ satisfies the equations [3, page 93]:
Pk,n = P0,nk!pk, 0<k<n, PKn = P0,nnnn\ (pn)k, k>n,
P0~,1 = rk=0 1k\pk + £k>n nnn! (pn)k. It is not difficult to obtain the following equality by simple algebraic calculations: a(t) = Ek>o V-Pk,nmin(k, n) = up = X. Therefore, Theorem 1 follows the validity of Theorem 2.
Remark. Using the scheme of the proof of Theorem 2, it is possible to extend the results to output flows of systems with limited queue, with priority service, with unreliable servers [3, $$].
Theorem 3. When the X> nn inequality is executed, the output flow Tn in the system An is Poisson with intensity a(t)^n^, t ^ We introduce independent random variables un(t), vn(t), having Poisson distributions with parameters Xt, ^t, respectively. It is easy to establish that with probability unit the inequality xn(t) > wn(t) = un(t) — vn(t) is valid, hence the following inequalities are fulfilled:
P(xn(t) <n)< P(wn(t) <n) = P(wn(t) — Mwn(t) <n — Mwn(t)) =
= P(Mwn(t) — wn(t) > Mwn(t) — n)< P(\Mwn(t) — wn(t)\ > Mwn(t) — n), where Mwn(t) = (X — n)t > 0, t> 0, Dwn(t) = (X + n)t. Assume that (X — n)t — n> 0, then, due to Chebyshev's inequality, we have:
P(\Mwn(t) — wn(t)\ > Mwn(t) — n)< Dwn(t)(Mwn(t) — n)2 = (X + v)t((X — n)t — n)2 ^ 0,
t ^
Finally we get the ratio:
P(xn(t) < n) < (X + n)t((x — n)t — n)2^0, t^ and then £0<k<n Pkin (t) ^ 0, t ^ x>. It follows that the limit ratio is met:
a(t) = £0<k<n Pk,n(t)k^ + (1 — £o<k<n Pk,n(t))w ^n^ t^ From Theorem 1 and the last relation we obtain the statement of Theorem 3.
2 Poisson flows in stationary queuing networks
Consider an open queuing network (Jackson network) S with a Poisson input flow of intensity X0, consisting of a finite number of nodes k = 0,1, ...,m with exponentially distributed service times. The dynamics of the movement of customers in the network is set by the route matrix 0 = ||0y||y=o, where is the probability of customer transition after service in the i-th node to y'-th node, 800 = 0, where the node 0 is an external source and at the same time a drain for customers leaving the network. The i node contains < ^ servers, the service time of which has an exponential distribution with the parameter i = 1,... ,m.
Assume that route matrix 0 = | |0y ||y=0 is indecomposable, i.e.
v i, je {0.....m} 3 ii.....ir E {0.....m}: 6iU > 0,9^ > 0.....9irJ > 0.
Tsitsiashvili G., Osipova M.
GENERALIZATION AND EXTENSION OF BURKE THEOREM
RT&A, No 1 (48) Volume 13, March 2018
Then for a fixed X0 > 0, the system of linear algebraic equations for intensities of fluxes coming from nodes of S
Xk = Ao&o,k + Y?t=i ^t6t,k, k = 1,...,m (1)
has the only solution (X1, ...,Xm) with X1 > 0, ...,Xm > 0, [4, p. 13].
The system (1) is called the system of balance relations and plays an important role in the formulation and the proof of the product Jackson theorem [5], widely used in queuing theory. If Xl < ll^l, i = 1,...,m, then the discrete Markov process (n1(t),...,nm(t)), t > 0, describing the number of customers in the network nodes has a limiting distribution Ps(k1, ...,km), independent of initial conditions and representable in the form Ps(k1,...,km) = U.t=1 Pl(^l), where Pi(ki) is the limiting distribution of the number of customers in a stand-alone i - channel queuing system with Poisson input flow of intensity Xl, i = 1,..., m.
In [6] network S is mapped to a directed graph G with edges corresponding to positive elements of the route matrix. Let's call the vertex set U £ {0,1, ...,m} irrevocable if from any node not included in U, there is no edge to the node belonging to U. Then all flows passing through the edges from the node set U to the node set {0,1, ...,m}\U, are independent and Poisson.
Theorem 4. Flow Tsl, coming out of node i of open queuing network S, with stationary process (n1(t), ...,nm(t)), t > 0, is Poisson with intensity Xl, i = 1, ...,m. Indeed, the points of the flow Tsl, exiting the i, node are the moments of jumps down the nl(t) component of the discrete Markov process (n1(t), ...,nm(t), t > 0. Hence the flow TS satisfies the condition a). Conditions b), c) are checked similarly to the proof of Theorem 1. Note that the limit probability that the node contains kl of customers is Pl(kl), and the flow rate TS is Xl, i = 1, ...,m.
Theorem 5. Flows T^, i = 1, ...,m, are independent. From Theorem 4 and independence of stationary random variables nj(t), j = 1,..., m, it follows that the union Ts =uj=1 tj of flows leaving the nodes of open queuing network S is also Poisson flow with intensity = Yj=1 ^J. And each point of the combined flow Ts belongs to the flow TS with probability pl = XlX-L.
Lemma 1. Let A = {0 < ^ < t2 < — } is a Poisson flow of intensity Xs, each point of which, regardless of other points with probability pl becomes a flow Al point, i = 1,..., m. Then flows A1,..., Am are Poisson with intensities Xsp1, ..,Xspm and independent. Without limitation of generality it is enough to limit ourselves to the case of m = 2. Take an arbitrary segment [t,t + T], 0 < t, 0 <T and denote n, n1, n2 the number of flow points A, A1, A2 on this segment, respectively. Calculate the probability
P(n1 = k.1, n2 = k^) = P(n = k.1 + k2)c£l+k2p*1p22
= e-XT(XT)kl+k2(k1 + k2)\ (k1 + k2)\k^. k2\pk11pk22 = = e-XTpi(XTp1)kik1. ■ e-XTp2(XTp2)k2k2. = P(n1 = k1) ■ P(n2 = k2). (2)
Let now segments [t(1),t(1) + T(1)], ...,[t(k),t(k) + T(k)] of the time axis don't intersect. Similarly to (2) we prove the independence of the random vectors (n^,... ,n(k)), (n(1), ...,n(k)) regarding the respective intervals, and independence of their components. Hence the flows Tsl, i = 1,...,m, are independent.
Remark. Theorems 4, 5 enhance the results of the article [6], removing restrictions on the independent Poisson flows considered in it.
Consider now a closed queueing network S, consisting of a finite number of nodes i = 1,...,m. The i node contains ^ < ro servers, the service time on which has an exponential distribution with the parameter , i = 1, ...,m. A finite number N of customers move along network S. The dynamics of the customers movement in the network is specified by the matrix 0 = h^i.j I \ij=1, where 9lsj is the probability of transition after service of customer in the ith node to y'-th one.
Let the route matrix 0 be indecomposable, i.e.
V i, je{1.....m}3 i1.....irE{1.....m}: > 0,9^ > 0.....> 0.
Then for a fixed 1 > 0, the system of linear algebraic equations
Xk=Zm=1Xt9t,k, k = 1.....m (3)
Tsitsiashvili G., Osipova M. RT&A, No 1 (48) GENERALIZATION AND EXTENSION OF BURKE THEOREM_Volume 13, March 2018
has a unique solution of (A1, ...,Am) with A1 > 0, ...,Am > 0, [4, p. 13].
For a closed queueing network S with N customers discrete Markov process (n1(t),... ,nm(t)), t > 0, describing the number of customers in the network nodes has a limit distribution of P^(k1,..., km), independent of the initial conditions and presented in the form
Ps(ki.....km) = n?=1 Pi(ki)£kl.....km: kl+...+km=N U?=1 Pi(ki), k_1 + - + km = N.
Hence, the stationary probability nt(ki) that in a node i of the network S there is kt customers satisfies the equality
ni(ki) = 'Ekj,1<j*i<m,Y,1<j^l<mkj=N-kl P's(i1, .■■ ,km), ki = 0,...,N. Theorem 6. The flow T±, leaving the i node of the closed queueing network S with the total number of customers N, being in a stationary state, is Poisson with intensity EN^-l min(ki, li)nini(ki), i = 1, ...,m. Indeed, the points of the flow T±, exiting the node i, are the moments of jumps down the components n^t) of the discrete Markov process (nL(t), ...,nm(t)), t > 0. Consequently, the flow t^ satisfies condition a). Conditions b), c) are proved similarly to the proof of Theorem 1. Example. Theorem 6 allows us to consider flows in systems backup with recovery. For example, for the simplest S system, a restore reservation consisting of one workstation (work phase 1), one repair location (repair phase 2), and one item. Let the random time before the failure of the item in the workplace has an exponential distribution with the parameter a, the random time to restore the item in the repair location has an exponential distribution with the parameter p. Denote nL(t) the number of elements in the working phase, n2 (t) - the number of elements in the repair phase of s. Then nL(1) = 0,1, n2(t) = 0,1, nL(t) + n2(t) = n = 1, and therefore the route matrix of the system on has the form d = Hd^ \ ^j^, where 9i L = d222 = 0, di22 = d2-1 = 1. Then equalities are just
Ps(1,0) = Pa + p, Ps(0,1) = aa + p, ^(0,0) = ^(1,1) = 0, nL = pa + p, n2 = aa + p, vl = v2 = apa + p,
where vl, v2 are the intensities of stationary Poisson flows leaving phases 1, 2, respectively.
Partially supported by Far Eastern Branch of Russian Academy Sciences, Program "Far East" (project 18-5-044).
References
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