The Effect of a MAP Flow on Performance Measures

D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

DCCN 2018 17-21 September 2018

UDC: 519.217

The Effect of a M AP Flow on Performance Measures of Controllable Queueing System with Heterogeneous Servers in a Random Environment D. Efrosinin1,2 , A. Krishnamoorthy3 , V. Vishnevskiy1 , D. Kozyrev1,4 1

4

V.A. Trapeznikov Institute of Control Sciences of RAS, Profsoyuznaya st., 65, 117997 Moscow, Russia 2 Johannes Kepler University, Altenbergerstrasse, 69, 4040 Linz, Austria 3 Department of Mathematics, CMS College, Kottayam-686001, India

Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russia [email protected], [email protected], [email protected], [email protected]

Abstract In this paper we study a M AP/M/2 queueing system with heterogeneous servers. The system operates in a Markovian random environment which modulated service rates of faster server. The allocation of customers between the servers proceeds according to a heuristic FFS-policy (Fastes Free Server) and to an optimal policy which minimizes the average number of customers in the system. It is shown that the correlated arrivals may have different impact on the average number of customers in the system and on other performance measures dependent on the value of correlation, properties of the random environment and type of the allocation policy. Keywords: Markovian arrival process, heterogeneous servers, random environment, optimal threshold policy, FFS-policy

1. Introduction A modern wireless communication system can be supplied by a heterogeneous mix of multiple data transmission links based on equipment, hardware configuration and technology of several generations and operating according to different physical principles. Hence the links may differ, especially in terms of data transmission speeds, The work has been carried out with the partial financial support from the Russian Science Foundation and the DST (India) (grant No. 16-49-02021) within the joint research project of the V.A.Trapeznikov Institute of Control Sciences of the Russian Academy of Sciences and the CMS College of Kottayam, India. This work was funded by the Russian Foundation for Basic Research, Project No. 16-37-60072 mol a dk.

D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

DCCN 2018 17-21 September 2018

reliability and availability. The complementary properties of different links motivate engineers to design hybrid communications systems. The possible example of such a system is a Radio Frequency/Free Space Optic (RF/FSO) hybrid channel that combines the advantages of both types of links [8, 17, 18]. The capacity of RF link is constrained by limits to link throughputs on the order of 10s of Mbps and distances of 10s of meter. It does not scale well with increasing number of nodes in the system due to the interference between concurrent transmission from neighbouring nodes. But the link availability can be maintained under most weather conditions. On the contrary, the commercial FSO currently provide throughputs of several Gbps with link distances of a kilometer or more. However, one of the major limitations of FSO technology is the need for optical links to maintain line-of-sight (LOS) and the FSO link availability can be further limited by adverse weather conditions like fogs and heavy snowfalls. Thus the combination of these two links seems to be a natural way to solve the capacity problem of RF and the problem of the reduced availability of the FSO due to the atmospheric effects. Obviously, the functioning of the hybrid channel with two types of unequal links can be modelled by virtue of the queueing system with two heterogeneous servers. The service rates of the faster server (can be considered as FSO link) are modulated by a random environment describing different weather conditions, while the slower one (RF link) does not require LOS condition and hence is independent on changing the states of random environment. To model the correlated inter-arrival times for the arrived customers we use Markovian arrival process (MAP) which is a convenient tool to describe renewal and non-renewal arrivals and hence is more appropriate for real situations. A discussion of a MAP in more detail can be found in [7, 10]. There are many papers dedicated to single-server queueing models with MAP flows, see e.g. [1, 16]. The multi-server queueing systems with MAP arrivals were studied mostly under the assumption of servers’ homogeneity, see e.g. [3, 4]. The M AP/M/2/K queueing system with finite buffer, two heterogeneous servers and RSS (Random Server Selection) allocation policy was analysed in [2]. The dynamic optimization problems for the queues of the type M AP/P H/K were studied in [6], where threshold and monotonicity properties of the optimal allocation policy were numerically confirmed. A number of papers study queueing systems in a random environment. Some special cases and literature overview for this field can be found in [5, 9]. To the best of our knowledge, there is no literature available on the performance analysis for multi-server controllable queues with MAP arrivals operating in random environment. Our analysis of the queueing system M AP/M/2 with heterogeneous servers and Markovian environment includes the following contributions: (a) The model is studied as a quasi-birth-and-death (QBD) process for the heuristic FFS-policy with specified stability condition.

D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

DCCN 2018 17-21 September 2018

(b) We develop the dynamic programming equations to evaluate the optimal policy which minimizes the long-run average number of customers in the system. It is shown that for the slower server exists an optimal threshold policy that depends on the queue length, on the state of the arrival process and on the state of the random environment. (c) The dynamic programming equations are used to calculate other performance measures for the given control policy. (d) We show that increasing of correlation of the inter-arrival times results in increasing of the average number of customers in the system and decreasing of the system utilization. The optimal policy can be more superior comparing to the FFS-policy even in case of higher correlation by high heterogeneity of servers. The rest of the paper is organized as follows. In Section 2, we describe the mathematical model and give a brief presentation of the QBD process for the FFSpolicy and stationary distribution of the system state. In Section 3, we develop the dynamic programming equations used for calculation of the optimal control policy. The algorithm for evaluation of the main performance measures is proposed in Section 4. In Sections 5, numerical illustrations are added to visualize the effect of some system parameters and control policies on the average number of customers in the system and other performance measures. Finally, potential further research ideas are presented in Section 6. For use in sequel, let e(r), ej (r) and Ir denote, respectively, the (column) vector of dimension r consisting of 1’s, column vector of dimension r with 1 in the jth position beginning from 0th and 0 elsewhere, and an identity matrix of dimension r. When there is no need to emphasize the dimension of these vectors we will suppress the suffix. Thus, e will denote a column vector of 1’s of appropriate dimension. The notation ”>” appearing in a vector will stand for the vector transpose. The notations ⊗ and ⊕ stand respectively for the Kronecker product and Kronecker sum of matrices. For more details on Kronecker products and sums, we refer the reader to [11]. 2. Mathematical model In this paper, we study a M AP/M/2 queueing system with heterogeneous servers where first server has larger service rate as the second one. A Markov arrival process (MAP) is defined by two transition matrices, D0 and D1 of dimension l, where ΛM = D0 + D1 is the generator of the continuous time Markov chain {M (t)}t≥0 associated with an arrival process with a state space EM = {1, . . . , l}, i.e. ΛM e(l) = 0. Here D0 = [ξij ]1≤i,j≤l , contains transitions without arrival and D1 = [νij ]1≤i,j≤l includes transition rates accompanying with a new arrival. Note

D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

DCCN 2018 17-21 September 2018

P  P that ξii = −ξi = − ν + ξ j ij j6=i ij . The average arrival rate λ = αD1 e(l), where α is the solution of the linear system αΛM = 0, αe(l) = 1. The service rate of the server 1 is modulated by a random environment associated with a irreducible continuous time Markov chain {N (t)}t≥0 with a finite state space P EN = {1, . . . , r} and infinitesimal generator ΛN = [γij ]1≤i,j≤r , γii = −γi = − j6=i γij . The service rate of the slower server is independent of the state of the external environment and is denoted by µ2 . If the external environment {N (t)}t≥0 is in state n ∈ EN \ {r}, then the above queueing system behaves as M AP/M (µ1n ), M (µ2 )/2 queue, otherwise it behaves as M AP/M (µ2 )/1 until the process leaves state r. We assume that with increasing of n the service rates µ1n decrease, i.e. µ1n ≥ µ1n+1 ≥ µ2 . Moreover, state r corresponds to the worst external condition that leads to the failure of the first server. The empty failed server becomes unavailable for service, i.e. µ1r = 0. The customer being served at the moment when failure occurs leaves this server and joins a queue or occupies an idle second server. The repair of the first server is connected with a transition of the external environment to one of states of the set EN \ {r} and can be accompanied by allocation of customer from the queue to the first server. It is easy to verify that the average service rate is defined as µ = pT e(r) + µ2 , where T = diag{µ11 , . . . , µ1r−1 , 0} and p is a stationary distribution of the process {N (t)}t≥0 calculated through solving the linear system pΛN = 0, pe(r) = 1. Heterogeneous servers require the definition of some control mechanism or policy to allocate the customers between the servers. If FFS-policy (Fastest Free Server) is selected, i.e. when the fastest free server must be chosen whenever a customer is allocated between the servers, the described system can be modelled by a quasibirth-and-death (QBD) process. Let Q(t) ∈ EQ = N0 denote the number of waiting customers at time t, Sj (t) ∈ ES = {0, 1} the state of the server j at time t, where ( 0 if server j is idle Sj (t) = 1 if server j is busy. The system states at time t are described by a multi-dimensional continuous time Markov chain {X(t)}t≥0 = {Q(t), S1 (t), S2 (t), M (t), N (t)}t≥0 .

(1)

with a state space EX defined as EX = E(0, 0, 0) ∪ E(0, 1, 0) ∪ E(0, 0, 1)

∞ [ q=0

E(q, 1, 1),

(2)

D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

DCCN 2018 17-21 September 2018

where E(0, 0, 0) = {(0, 0, 0, m, n) : m ∈ EM , n ∈ EN }, E(0, 1, 0) = {(0, 1, 0, m, n) : m ∈ EM , n ∈ EN \ {r}, E(0, 0, 1) = {(0, 0, 1, m, n) : m ∈ EM , n ∈ EN }, E(q, 1, 1) = {(q, 1, 1, m, n) : m ∈ EM , n ∈ EN \ {r}} ∪ {(q + 1, 0, 1, m, r) : m ∈ EM }. The Markov chain (1) is of the QBD type with a infinitesimal generator 

ΛX

B00 B01 B10 B11  B20 0   0 B31  = 0 0   0 0   0 0  .. .

B02 0 0 0 0 B12 B13 0 0 0 B22 A0 0 0 0 B32 A1 A0 0 0 0 A2 A1 A0 0 0 0 A2 A1 A0 0 0 0 A2 A1 .. .. . .

 ... . . .  . . .  . . .  . . . .   . . . . . .  .. .

(3)

The block matrices in ΛX are of the form: B00 = ΛN ⊕ D0 , B11 = (Λ∗N − T ∗ ) ⊕ D0 , B22 = (ΛN − µ2 Ir ) ⊕ D0   Ir−1 ⊗ D1 B01 = , B12 = (α1r , . . . , αr−1r )> ⊗ e> l−1 (l) ⊗ Il , 0l×l(r−1)    ∗ 
µ2 I(r−1)l T 0 B10 = T ∗ 0(r−1)l×l ⊗ Il , B20 = µ2 Irl , B31 = , B32 = ⊗ Il , 0l×(r−1)l 0 µ2 the matrix Λ∗N is obtained from ΛN by deleting the last row and column, and T ∗ = diag{µ11 , . . . , µ1r−1 }. A homogeneous three-diagonal part of the generator ΛX is defined by matrices A0 , A1 and A2 , A0 = Ir ⊗ D1 , A1 = (ΛN − T − µ2 Ir ) ⊕ D0 , A2 = (T + µ2 Ir ) ⊗ Il , respectively for the transitions from level q − 1 to q due to arrivals, for transitions inside the subset q and for transitions from level q + 1 to q due to service completions. Theorem 1. The Markov chain {X(t)}t≥0 under FFS-policy is ergodic if λ 0, a ∈ A(x − e0 − e1 ),      y = x − e2 , s2 (x) = 1, q(x) = 0, µ2 , λxy (a) = µ2 , y = x − e0 − e2 , s2 (x) = 1, q(x) > 0, a ∈ A(x − e0 − e2 ),    γn(x)i , y = x + (i − n(x))e4 , i ∈ EN \ {n(x), r},      γn(x)r , y = x + (r − n(x))e4 + ea , a ∈ A(x + (r − n(x))e4 ),      γri , y = x + (i − r)e4 , i ∈ EN \ {r}, q(x) = 0, n(x) = r,      γri , y = x + (i − r)e4 − e0 + ea , i ∈ EN \ {r}, q(x) > 0, n(x) = r,     a ∈ A(x + (i − r)e4 − e0 ). – c(x, a) is an immediate cost in state x ∈ E under control action a ∈ A(x), c(x, a) = c(x) = q(x) + s1 (x) + s2 (x), which defines the number of customers in state x and is independent of a. A controller or decision maker chooses an action according to the following decision rule which will refer to as stationary policy. Definition 1. A stationary policy is a function f : E → A(x) which prescribes a selection of a control action a ∈ A(x) whenever the process {X(t)}t≥0 is in state

D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

DCCN 2018 17-21 September 2018

x ∈ E just after an arrival, just after a service completion if q(x) > 0, just after a failure or just after a repair. Any other transition cannot impute a selection of some action. Remark 1. For the infinite state Markov decision process with unbounded costs the existence of an optimal stationary policy and convergence of the sequence of improved policies to the optimal one under the average cost criterion must be verified. To do it we employ the main theorem of Sennott [15], by consecutive checking whether our model satisfies Assumptions 1,2,3, and 3∗ of [15]. For any fixed stationary policy f we wish to guarantee that the process {X(t)}t≥0 f is an irreducible, positive recurrent Markov chain with a state space EX and inf f finitesimal generator ΛX = [λxy (a)]. As it is known, for ergodic Markov chains with costs the long-run average cost per unit of time for the policy f coincides with corresponding assemble average, X 1 c(y)πyf , (11) g f = lim sup V f (x, t) = t t→∞ f y∈E

Rt where V f (x, t) = Ef [ 0 (Q(t) + S1 (t) + S2 (t))dt|X(0) = x] denotes the total average cost up to time t given initial state is x and πyf = Pf [X(t) = y] denotes a stationary state probability of the process under given policy f . The policy f ∗ is said to be optimal when g ∗ = inf g f . f

(12)

The ergodicity condition for the Markov chain {X(t)}t≥0 can be obtained for the process operating under a heuristic FFS-policy (Fastest Free Server) which prescribes to allocate the customers to the fastest idle server. This model can be analysed as a quasi-birth-and-death (QBD) process with tri-diagonal block infinitesimal generator. One fruitful approach to finding optimal policy f ∗ is through solving average cost optimality equation Bv(x) = v(x) + g,

(13)

where B is a dynamic programming operator acting on value function v : E → R which indicates a transient effect of an initial state x to the total average cost and satisfies an asymptotic relation V f (x, t) = g f t + v f (x) + o(1), x ∈ E, t → ∞.

(14)

The optimality equation allows to construct a sequence of improved policies until the average cost optimal is reached. The functions v f and g f further in the paper will be denoted by v and g without upper index f .

D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

DCCN 2018 17-21 September 2018

Theorem 3. The dynamic programming operator B is defined as follows h 1 Bv(x) = c(x) (15) ξm(x) + µ1n(x) s1 (x) + µ2 s2 (x) + γn(x) X X ξm(x)i v(x + (i − m(x))e3 ) + νm(x)i T v(x + (i − m(x))e3 ) + i∈EM

i∈EM \{m(x)}

+ µ1n(x) s1 (x)[v(x − e1 )1{q(x)=0} + T v(x − e0 − e1 )1{q(x)>0} ] + µ2 s2 (x)[v(x − e2 )1{q(x)=0} + T v(x − e0 − e2 )1{q(x)>0} ] X + γn(x)i v(x + (i − n(x))e4 ) i∈EN \{n(x),r}

+ γn(x)r [s1 (x)T v(x + (r − n(x))e4 ) + (1 − s1 (x))v(x + (r − n(x))e4 )]1{n(x)6=r} X + γri [v(x + (i − r)e4 )1{q(x)=0} i∈EN \{r}

i + T v(x + (i − r)e4 − e0 )1{q(x)>0} ]1{n(x)=r} , where operator T stands for decision making at decision epoch and is of the form T v(x) = min v(x + ea ).

(16)

a∈A(x)

Proof. According to [14] in general case we have io n 1 h X c(x) + λxy (a)v(y) . Bv(x) = min a λx (a) y6=x

Evaluating these equations for analysed queueing system and taking into account the transition rates of the specified Markov decision model we get (15), where the term c(x) is a number of customers in state x ∈ E, the next two terms represent the changing of the state of MAP accompanying with an arrival with a rate νm(x)i and without arrival with a rate ξm(x)i . The next two terms represent transitions due to service completions with rates µ1n(x) and µ2 at first and second server respectively. The last three terms represent transitions of the random environment with rates γn(x)i . At a decision epoch a new state after transition is obtained using relation (16).  4. Performance measures The dynamic programming approach can be used not only for evaluation of the optimal policy (OP) and the optimized average number of customers in the

D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

DCCN 2018 17-21 September 2018

¯ . By changing the immediate cost function c(x) we can calculate other system N average performance measures as well. To calculate the stationary state probability πx = P[X(t) = x] the immediate cost must be set to c(x) = 1 and c(y) = 0 for y = 6 x. Corollary 1. Utilization of the first server ¯1 = P[S1 (t) = 1] = U

∞ X 1 X l X r X

π(q,1,s2 ,m,n) =

q=0 s2 =0 m=1 n=1

X

c(x)πx ,

x∈E

where c(q, 1, s2 , m, n) = 1, q ∈ EQ , s2 ∈ ES , m ∈ EM , n ∈ EN and c(x) = 0 otherwise. Utilization of the second server ¯2 = P[S2 (t) = 1] = U

∞ hX l X r X q=0

π(q,0,1,m,n) +

m=1 n=1

l X r−1 X

i X π(q,1,1,m,n) = c(x)πx ,

m=1 n=1

x∈E

where c(q, 0, 1, m, n) = 1, q ∈ EQ , s2 ∈ ES , m ∈ EM , n ∈ EN , c(q, 1, 1, m, n) = 1, q ∈ EQ , s2 ∈ ES , m ∈ EM , n ∈ EN \ {r} and c(x) = 0 otherwise. Utilization of the system ¯ = P[Q(t) + S1 (t) + S2 (t) > 0] = 1 − U

l X r X

π(0,0,0,m,n) = 1 −

m=1 n=1

X

c(x)πx ,

x∈E

where c(0, 0, 0, m, n) = 1, m ∈ EM , n ∈ EN and c(x) = 0 otherwise. Average number of customers in the queue ¯= Q

∞ h X 1 X l X r X q=0

qπ(q,0,s2 ,m,n) +

s2 =0 m=1 n=1

1 X l X r−1 X s2 =0 m=1 n=1

i X c(x)πx , qπ(q,1,s2 ,m,n) = x∈E

where c(x) = q(x), x ∈ E. Average number of busy servers C¯ =

2 X

¯j = N ¯ − Q. ¯ U

j=1

The infinite buffer queueing system is approximated by a finite buffer equivalent system. The buffer size B satisfies the condition B>

log ε λ ,ρ= , log ρ µ

obtained for M/M/1 queueing system, where the sum of stationary state probabilities P∞ π i=B i < ε. For bounded puffer size q ≤ B the size of the set space |E| =

D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

DCCN 2018 17-21 September 2018

2l(B + 1)(2r − 1). Further we convert the five dimensional state space E of the Markov decision process to a one-dimensional equivalent state space N0 , ∆ : E → N0 , for state x = (q, s1 , s2 , m, n) ∈ E ∆(x) = q(x)2l(2r − 1) + 4l(n(x) − 1)

(17)

+ 2(m(x) − 1) + s2 (x) + (s1 (x) + s2 (x) + 2(m(x) − 1))1{n(x)0} ]1{n(x)=r} . Step 3. Policy improvement fk+1 (x) = arg min vk (x + ea ), x ∈ E. a∈A(x)

Step 4. Check the convergence. If fk+1 (x) = fk (x) for all x ∈ E, then f ∗ (x) = fk+1 (x) is an optimal policy, otherwise k → k + 1 and go to Step 2. Step 5. For evaluated optimal policy f ∗ (x), x ∈ EX , calculate performance measures using equations in Step 2 replacing the number of customers in state x by appropriate function c(x) defined for each characteristic in Corollary 1 . Example 1. Consider the system M AP/M/2, where matrices D0 , D1 and ΛN are taken from Case 4B in Section 5. The service rates are: µ11 = 10, µ12 = 4, µ2 = 1. Table 1 gives optimal control actions for allocation of customer in a certain state in form of a control matrix. Here the optimal policy f ∗ is defined completely through the sequence of 9 optimal threshold levels ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ q∗ = (q11 , q21 , q31 , q12 , q22 , q32 , q13 , q23 , q33 ) = (4, 13, 13, 1, 2, 3, 3, 6, 6)

D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

System state x (s1 , s2 , m, n) (0, s2 , m, n) n < r (1, 0, 1, 1) (1, 0, 2, 1) (1, 0, 3, 1) (1, 0, 1, 2) (1, 0, 2, 2) (1, 0, 3, 2) (0, 0, 1, 3) (0, 0, 2, 3) (0, 0, 3, 3) (0, 1, m, 3) (1, 1, m, n) n < r

0 1 0 0 0 2 0 0 0 0 0 0 0

1 1 0 0 0 2 2 0 0 0 0 0 0

2 1 0 0 0 2 2 2 2 0 0 0 0

3 1 0 0 0 2 2 2 2 0 0 0 0

4 1 2 0 0 2 2 2 2 0 0 0 0

5 1 2 0 0 2 2 2 2 2 2 0 0

6 1 2 0 0 2 2 2 2 2 2 0 0

Queue 7 1 2 0 0 2 2 2 2 2 2 0 0

length q(x) 8 9 10 1 1 1 2 2 2 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0

DCCN 2018 17-21 September 2018

11 1 2 0 0 2 2 2 2 2 2 0 0

12 1 2 2 2 2 2 2 2 2 2 0 0

13 1 2 2 2 2 2 2 2 2 2 0 0

14 1 2 2 2 2 2 2 2 2 2 0 0

... 1 2 2 2 2 2 2 2 2 2 0 0

Table 1. Control matrix of the system

respectively for the states (s1 , s2 , m, n) ∈{(1, 0, 1, 1), (1, 0, 2, 1), (1, 0, 3, 1), (1, 0, 1, 2), (1, 0, 2, 2), (1, 0, 3, 2), (0, 0, 1, 3), (0, 0, 2, 3), (0, 0, 3, 3)}. The policy f ∗ prescribes the usage of a faster server whenever it is free and operational and of a slower server if upon arrival to one of these states the number of customers in the queue exceeds the corresponding threshold level. The optimal value of the ¯ = 9.6318. Other performance average number of customers in the system is N ¯ = 8.6695, U ¯ = 0.5644. measures for the optimal threshold policy take the values: Q Numerical results confirm our expectation that for slower server exists an optimal threshold policy with a threshold level that depends on the queue length q(x), on the state of the arrival process m(x) and on the state of the random environment n(x). By fixed thresholds the model can by analysed as a QBD-process with three-diagonal block structure of the homogeneous part of the infinitesimal generator. But due to a large number of boundary states we prefer to use a dynamic programming approach which definitive has advantages since it allows to calculate the average performance measures together with optimal thresholds. In particular case, when all thresholds are set to be equal to 1, we get performance values of the system operating under the FFS-policy. 5. Numerical results We consider four different MAP flows defined by matrices D0 and D1 . The elements of these matrices were taken from [4]. The average rate of MAPs was fixed to λ = 5 and the service rate of the second server was fixed to µ2 = 1. The lag-1

D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

DCCN 2018 17-21 September 2018

correlation coefficient ρ of the inter-arrival times is computed by ρ=

λα(−D0 )−1 D1 (−D0 )−1 e(l) − 1 , c2v

where cv is a coefficient of variation given by cv = 2λα(−D0 )−1 e(l) − 1. For MAP {M (t)}t≥0 we consider four cases. Case 1: Stationary Poisson arrival stream with D0 = −5 and D1 = 5. In this case ρ = 0 and cv = 1. Case 2: MAP is defined by matrices     −13.3346 0.5886 0.6173 11.5469 0.3631 0.2187 −2.4466 0.4229  , D1 =  0.3842 0.8659 0.0809 D0 =  0.6927 0.6823 0.4144 −1.6354 0.2852 0.0425 0.2110 with ρ = 0.1 and cv = 2. Case 3: MAP is defined by matrices   −15.7327 0.6062 0.5924 −2.2897 0.4679  , D0 =  0.5178 0.5971 0.5653 −1.9597

  14.1502 0.3021 0.0818 D1 =  0.1071 1.0320 0.1649 0.0858 0.1979 0.5136

with ρ = 0.2 and cv = 2. Case 4: MAP is defined by matrices   −25.5398 0.3933 0.3612 −2.2322 0.2000  , D0 =  0.1452 0.2960 0.3874 −1.7526

  24.2421 0.4669 0.0763 D1 =  0.0341 1.6668 0.1861 0.0090 0.2555 0.8047

with ρ = 0.3 and cv = 2. For the random environment {N (t)}t≥0 we differ two cases with small and large repair rates:     −0.3 0.1 0.2 −0.3 0.1 0.2 Case A: ΛN =  0.1 −0.2 0.1  Case B: ΛN =  0.1 −0.2 0.1  0.5 0.3 −0.8 5.0 3.0 −8.0 Figures 1–3 illustrate respectively the effect of the correlation coefficient ρ on ¯, Q ¯ and U ¯ by varying service rates µ11 = µ12 ∈ [8.0, 20.0] for performance measures N Cases 1A–4A (the figures labelled by a“) and for Cases 1B–4B (the figures labelled ” by b“). Figures 1 and 2 show that as long as the correlation coefficient ρ increases ”

D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

DCCN 2018 17-21 September 2018

(a)

(b)

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Μ11

¯ versus µ11 = µ12 Fig. 1. The average number of customers in the system N (a)

(b)

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(b) 1.0

U

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D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

DCCN 2018 17-21 September 2018

the average number of customers in the system and in the queue increases at times. This fact reveals the importance and necessity to take into account the effect of correlation of inter-arrival times by design and forecast of the loading in queueing systems with non-Markovian arrival flows. Comparing the curves for optimal policy (OP) with ρ = 0.1 and ρ = 0.2 with the curves for fastest free server policy (FFS) with ρ = 0 we notice that OP can be more superior in performance against FFS also for correlated inter-arrival times if heterogeneity of servers is getting higher. In this case the OP can compensate increased correlation by more flexible allocation of customers between the servers. The effect of system parameters and control policy on system utilization is illustrated in Figure 3. We see that the FFS-policy is more sensitive to varying of correlation coefficient comparing to the OP. By increasing the arrival flow correlation the utilization of the system operating under FFS-policy decreases while for OP the curves are intersecting and locate quite close to each other. The fact that the system utilization decreases while the average number of customers increases together with increasing of the coefficient of correlation is not a contradiction. It can be explained by the observation that the stationary state distribution π of the system by high correlated arrivals exhibits a heavier tail comparing to low correlated flows. It leads to lower probabilities of the states with small number of customers in the system and to higher probabilities of the states with large number of customers for correlated flows and as a result – to higher values ¯ and Q. ¯ of N The influence of parameters of the random environment can be analysed using figures corresponding to Cases A and B. We notice that increasing of the repair rates in state r leads to a reduction of number of customers in the system and in the queue. Also we notice the decreasing of the system utilization if the random environment has higher transition rates to states in EN \ {r}, where the faster server becomes operational again. 6. Conclusion We have studied a M AP/M/2 queueing system with heterogeneous servers operating in a Markovian random environment. For fixed stationary control policy used for allocation the customers between the servers the model can be formulated as a QBD-process. For optimal policy under average cost criterion due to a large number of boundary states the dynamic programming approach seems to be more appropriate to provide performance analysis and study the effect of correlation of the inter-arrival times on performance measures. There are some potential developments of this topic, e.g. the average busy period analysis which can be performed using dynamic programming methodology and estimation of optimal thresholds as function

D. Efrosinin, A. Krishnamoorthy, et al. The Effect of a M AP Flow on Performance Measures

DCCN 2018 17-21 September 2018

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