Performance Measures Computation for a Single Link Loss Network ...

Performance Measures Computation for a Single Link Loss Network with Unicast and Multicast Traffics Irina Gudkova and Olga Plaksina Peoples Friendship University of Russia, Telecommunication Systems Department, Ordzhonikidze str. 3, 115419 Moscow, Russia {igudkova,oplaksina}@sci.pfu.edu.ru http://www.telesys.pfu.edu.ru

Abstract. The resource sharing is a fundamental issue for multirate loss networks, specially for multicast traffic with a huge number of recipients. In this paper, we give a generalized mathematical model to compute exact values of performance measures for unicast and multicast multirate traffics sharing a single link. We propose a recursive algorithm to obtain the normalization constant and the main performance measures, including blocking probabilities and link utilization factor. Keywords: Multiservice loss networks, multicast, unicast, blocking probabilities, recursive algorithm.

1

Introduction

The resource sharing problem is a key problem in multiservice loss networks. Multirate unicast loss models have been widely studied. An effective recursive algorithm to calculate blocking probabilities was proposed in [4], [5]. A set of works was devoted to multicast traffic models including [2], [3], [7]. Two different users’ behavior models were studied. The first one is based upon the assumption that all multicast users are involved in a session from the moment they join it until the user initiated this session departs. The second one assumes that all users depart independently and the session is active if there is at least one multicast user. The first model was introduced in [7] where the recursive formula for the main performance measures was proposed. Nevertheless, it has a limitation as it allows to compute characteristics for the one multicast service only. The second model was studied by a number of researchers. In [3] the multiservice loss model was introduced and the exact blocking probabilities were deduced. Then, based on this work in [2] a recursive algorithm for a multicast single link network was derived. There was also proposed a convolution formula for a model with a mixture of unicast and multicast traffic; however the use of convolution led to significant computational costs. The model [7] was extended for the case of two traffic types in [6] where a recursive algorithm was proposed. S. Balandin et al. (Eds.): ruSMART/NEW2AN 2010, LNCS 6294, pp. 256–265, 2010. c Springer-Verlag Berlin Heidelberg 2010

Performance Measures Computation for a Single Link Loss Network

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In this paper, considering the results obtained in [6] and [2] we generalize a model of a single link loss network with unicast and multicast traffics independently of multicast users’ behavior. We present a new recursive algorithm to calculate a number of performance measures. Note that the proposed additional recursion enables to reduce computational time in comparison with [6]. The rest of this paper is organized as follows. In the next section, we present the mathematical model of a single link loss network with multicast and unicast traffics and its product solution form. The main performance measures are also described. In section 3, we present the recursive algorithm to calculate blocking probabilities and other characteristics. Then in section 4, an example is given to illustrate the use of the algorithm. A summary of this paper is discussed in section 5.

2 2.1

Model of a Multiservice Single Link Loss Network with Unicast and Multicast Traffics Single Link Model

We consider a single link multiservice network supporting unicast and multicast traffics. The link has C capacity units (c.u.) and supports K := {1, . . . , K} classes of unicast connections and M := {1, . . . , M } multicast services, as shown in Fig.1.

Fig. 1. Model of a multiservice single link loss network with unicast and multicast traffic

Class-k connections arrive according to independent Poisson process with rate νk and have requirements of dk c.u., ak is the offered load, k ∈ K. Multicast traffic has a bandwidth saving nature. Users requested the same service are served simultaneously without occupying additional network resources. Such service discipline we refer as “transparent”. Consider two transparent disciplines T1 and T2 corresponding to multicast users’ behavior models described in [7] and [2] respectively. User requests for service m arrive according to independent Poisson process with rate λm . A request for service m is blocked and

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lost if both of the following conditions are true: service m is not provided to any user via the link and there are less than bm c.u. free. Otherwise the request is accepted. For the discipline T1 session duration equals service initiator residency time that is exponentially distributed with mean μ−1 m . During this period of time newly arriving users requested the same service are accepted without blockings, but they do not affect session duration. When this session ends all users depart together with the session initiator. For the discipline T2 session duration could be increased by newly arriving requests for service m. Each user request has exponential residency time with mean μ−1 m . Session ends when the last user departs. Let denote ρm := λm μ−1 m , m ∈ M. 2.2

State Probabilities

Firstly, we consider unicast traffic and introduce a process describing it. Let Nk (t) be the number of k-class connections in the system at time t, then the process {N (t) := (N1 (t) , . . . , NK (t)) , t ≥ 0} represents the states of unicast connections. Let d (n) := k∈K dk nk be the number of c.u. occupied in state n := (n1 , . . . , nK ), nk ∈ {0, . . . ,C/dk } =: Nk . The state space is given by N :=

n ∈ × Nk : k∈K

d (n) ≤ C . State probabilities for the process N (t) are

well known [1], [4]: π (n) = G−1 (N )

ank k , nk !

n∈N .

(1)

k∈K

In this paper, notation G (A) stands for a normalization constant for the corresponding process distribution with the state space A. Then, consider multicast traffic. In [2] the link states are described through the number of multicast users; in [6] they are described through the states of multicast services. The last approach let us to construct a general model independently of users’ behavior. Let Ym (t) equals 1 in the case when service m is active at time t, and 0 otherwise. The process {Y (t) := (Y1 (t) , . . . , YM (t)) , t ≥ 0} represents the states of multicast services. Let b (y) := m∈M bm ym be the number of c.u. occupied in state y := (y , . . . , y ), y ∈ {0, 1}. The state space is given by 1 M m M Y := y ∈ {0, 1} : b (y) ≤ C . From the global equilibrium balance equations it can be shown that state probabilities for process Y (t) are given by αymm , y ∈ Y , (2) π (y) = G−1 (Y) m∈M

with parameter αm := λm wm , where wm is the mean session duration for service m and depends on service discipline. For disciplines T1 and T2 (see Appendix) we have: −1 μm , for discipline T1 , (3) wm = , (eρm − 1) λ−1 m for discipline T2 .


259

Finally, we consider the case of both unicast and multicast traffics sharing the link. The process {Z (t) := (N (t) , Y (t)) , t ≥ 0} represents the system states. Let c (z) := d (n) + b (y) , z ∈ N × Y (4) be the number of occupied c.u. in state z := (n, y). The system state space is given by Z := {z ∈ N × Y : c (z) ≤ C} . (5) The process Z (t) is a reversible Markov process with equilibrium distribution ank k π (z) = G−1 (Z) αymm , z ∈ Z , (6) nk ! k∈K

m∈M

where G (Z) = π −1 (0) is the normalization constant. 2.3

Performance Measures

The main model performance measures are the probability BkI := P z ∈ BkI II II that a := P z ∈ Bm that a k-class connection is blocked, the probability Bm user request for service m is blocked, the probability Hm := P {z ∈ Hm } that service m is not offered, but the link has enough c.u. to satisfy a user request for the service, the probability Fm := P {z ∈ Fm } that service m is offered via the link, where BkI := {z ∈ Z : II := {z ∈ Z : Bm

C − dk + 1 ≤ c (z) ≤ C} ,

k∈K,

(ym = 0) ∧ (C − bm + 1 ≤ c (z) ≤ C)} ,

Hm := {z ∈ Z :

(ym = 0) ∧ (0 ≤ c (z) ≤ C − bm )} ,

Fm := {z ∈ Z :

ym = 1} ,

m∈M, m∈M,

m∈M,

and the utilization factor of the link 1

c (z) π (z) . UTIL := C

(7) (8) (9) (10)

(11)

z∈Z

Calculating performance measures for links with large capacity serving a huge number of users leads to large state space dimension and significant computational costs. In the next section, we give a recursive algorithm to avoid difficulties in computing.

3 3.1

Recursive Algorithm for Performance Measures Computation State Space Partitioning

To derive a recursive algorithm we use the approach based on partitioning the system state space Z for the number of occupied c.u.: Z=

C c=0

Z (c) ,

Z (c) := {z ∈ Z :

c (z) = c} , c = 0, . . . , C .

(12)

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The notation

Ai in this paper denotes that sets Ai are mutually disjoint.

i∈I

Then, for every service m subset Z (c) could be partitioned in two subsets according to its state – whether service m is active or not: Z (c) = {z ∈ Z (c) : ym = 0} m ∈ M, c = 0, . . . , C .

{z ∈ Z (c) :

ym = 1} ,

(13)

The first one subset we denote as Z−m (c) := {z ∈ Z (c) :

m ∈ M, c = 0, . . . , C .

ym = 0} ,

(14)

Now, sets (7) – (10) can be expressed through the defined subsets Z (c) and Z−m (c) as follows: BkI =

C

Z (c) ,

k∈K,

(15)

c=C−dk +1

II = Bm

C

Z−m (c) ,

m∈M,

(16)

c=C−bm +1

Hm =

C−b m

Z−m (c) , Fm =

c=0

C

Z−m (c) ,

m∈M.

(17)

c=0

II It is evident that Bm Hm Fm = Z. Taking into account that the link serves two different types of traffic, every subset Z (c) and Z−m (c) can be written down as a convolution Z (c) =

c

[Y (i) × N (c − i)] ,

c = 0, . . . , C ,

(18)

i=0

Z−m (c) =

c

[Y−m (i) × N (c − i)] ,

m ∈ M, c = 0, . . . , C ,

i=0

where N (c) := {n ∈ N : d (n) = c} , N =

C

N (c) ;

c=0

Y (c) := {y ∈ Y : b (y) = c} , Y =

C

Y (c) ,

c=0

Y−m (c) := {y ∈ Y (c) : ym = 0} .

(19)


3.2

261

Recursive Formula for Multicast Traffic

Performance measures for unicast connections can be calculated by the well known Kaufman–Roberts recursive formula [4], [5] ⎧ 0 c < 0, ⎪ ⎪ ⎨ 1, c = 0, h (c) = 1

(20) ⎪ a d h (c − d ) , c = 1, . . . , C , ⎪ k k k ⎩c k∈K

where h (c) is the unnormalized probability that exactly c c.u. are occupied. The recursive solution (20) cannot be applied for multicast traffic. Other solution was proposed separately for disciplines T1 and T2 in [7] and [2] respectively. Here we generalize this approach. Let ym := (y1 , . . . , ym ) be the m-dimension vector of the first m multicast m b y be the number of c.u. occupied services states; then let b (ym ) := i=1 i i by the first m multicast services in the state ym . The set of all possible states ym when the number of occupied c.u. is equal to c we denote as Y (c, m) := {ym : y ∈ Y, b (ym ) = c}. Then ⎧ ∅, c < 0, m = 0, 1, . . . , M, ⎪ ⎪ ⎪ ⎪ c = 0, m = 0, 1, . . . , M, ⎨ {0} , c = 1, . . . , C, m = 0, Y (c, m) = ∅, ⎪ ⎪ ⎪ Y (c, m − 1) × {0} ⎪ c = 1, . . . , C, m = 1, . . . , M, ⎩ Y (c − bm , m − 1) × {1} , (21) where notation A1 × A2 denotes the product set of sets A1 and A2 . The last line in the formula (21) goes from Y (c, m) = {ym ∈ Y (c, m) : ym = 0} {ym ∈ Y (c, m) : ym = 1} . (22) Note that Y (c, M ) = Y (c) and it let us calculating blocking probabilities for unicast traffic. For multicast traffic we need another recursion for sets Y−m (c) that has the following form: ⎧ c