Cybernetics and Systems Analysis, Vol. 47, No. 4, July, 2011
MULTIDIMENSIONAL ERLANG MODEL WITH RANDOMIZED CALL ADMISSION CONTROL STRATEGY AND ITS APPLICATION IN COMMUNICATION NETWORKS A. Z. Melikova and L. A. Ponomarenkob
UDC 519.872
Abstract. An analytical method is proposed for the analysis of a multidimensional Erlang model with randomized call admission control strategy. The applications of this model in both wireless multiservice cellular networks and integrated voice/data networks are demonstrated. Keywords: multidimensional Erlang model, randomized call admission control strategy, quality of service, calculation method, communication networks. INTRODUCTION Erlang’s model [1] has been widely used for more than hundred years for the analysis of various engineering service systems. Its scientific value has essentially increased after the Kovalenko classical result [2]: the stationary distribution of this model has been proved to be invariant with respect to the distribution function of holding time for fixed mean value. This was confirmed later by other authors (see, for example, [3] and the bibliography therein). It is especially expedient to apply this model in communication networks [4]. Since modern networks are multiservice, models with heterogeneous calls, i.e., multidimensional Erlang models, are actively investigated. Multi-velocity Erlang models, a more difficult class of multidimensional models, are also intensively analyzed. See [5, 6] and the monographs [7–9] for a detailed list of references on these models. The quality of service (QoS) in multidimensional Erlang models substantially depends on the call admission control (CAC) strategy for heterogeneous calls. These strategies are usually introduced to satisfy the set of constraints on the QoS of heterogeneous calls and on the utility of scarce resources (channels) of system. QoS criteria are sometimes economic indices related to the operation of the model under study. Some statements of problems of finding optimal CAC and methods to solve them can be found in [5–11]. In the paper, we propose an analytical approach to the calculation of QoS parameters for a multidimensional Erlang model with randomized CAC for different mean values of service times of heterogeneous calls. 1. DESCRIBING THE CAC STRATEGY AND CALCULATION METHOD Consider a queuing system described by a multidimensional Erlang model M K | M K | N | N , where calls of the ith type ( i-calls) have the arrival rate l i , and their average service time is m -1 i , i = 1, ¼, K . Randomized CAC strategy is used in this system. To this end, we define an K ´ N access matrix, whose elements specify the rules of accepting various calls depending on their type and the number of busy channels. We introduce the probabilities a i ( n ), i = 1,¼, K , n = 0, ¼, N - 1, a Institute of Cybernetics, National Academy of Sciences of Azerbaijan, Baku, Azerbaijan,
[email protected]. bInternational Scientific and Training Center of Information Technologies and Systems, National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine, Kyiv, Ukraine,
[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 117–124, July–August 2011. Original article submitted June 7, 2010.
606
1060-0396/11/4704-0606
©
2011 Springer Science+Business Media, Inc.
with a i ( N ) = 0 for any i = 1,¼, K . The value of a i ( n ) means the probability of accepting the i-call if the number of busy channels at the moment of its arrival is equal to n; with additional probability 1- a i ( n ) the call is lost. Note that a number of well-known CAC strategies can be obtained from this strategy in special cases. We will consider some of them. 1. If a i ( n ) = 1 for any i = 1,¼, K and n = 0, ¼, N - 1, we obtain complete sharing (CS) strategy. 2. Let the parameters a i ( n ) be defined as follows: ì 1 if n £ N i , a i (n ) = í î 0 otherwise
(1)
K ì ü S : = í m : mi = 0, N , å mi £ N ý . i =1 î þ
(2)
where 1 £ N K £ N K -1 £ ¼£ N 1 = N , i = 1, ¼, K . Then certain values of the parameters N i , i = 1,¼, K , in (1) yield different multiparameter CAC strategies based on channel reservations, which are widely used in communication networks (see [12–14] and the bibliography therein). The task is to find the main QoS parameters of the system: loss probabilities for calls of each type and channel utilization factor. Let us pass to the proposed problem solution method. Let the state of the system be described by a K-dimensional vector m = ( m1 , ¼, mK ), where mi specifies the number of i-calls in the system, i = 1,¼, K . Then the phase space (PS) of states of the corresponding K-dimensional Markov chain (MC) is defined by
Hereinafter, the notation a : = b means that a is defined by expression b. According to the introduced CAC strategy, non-negative elements of the Q-matrix of this chain are defined from the following relations: K
ì l i a i ( n ) if å mi = n, m¢ = m + e i , ï i =1 ï (3) q( m, m¢ ) = í mi m i if m¢ = m - e i , ï ïî 0 otherwise , where e i is the ith unit vector of the K-dimensional Euclidean space, i = 1,¼, K . For any positive values of parameters of the arrival traffic, all the states are communicating; hence, the system is ergodic. Denote the stationary probability of state m Î S by p( m ) . The unknown QoS parameters are defined by stationary probabilities of states. Let Pi be the loss probability for i-calls, i = 1,¼, K , and N av specify the average number of busy channels of the cell. Then using the PASTA property [15] we obtain that the above loss probabilities are defined by Pi : =
N
å (1- a i ( n )) å
n= 0
K ì ü where S n : = í m Î S : å mi = ný , n = 0, 1,... , N . i =1 î þ The average number of busy channels in the system is N
N av : = å n
p( m ),
(4)
mÎS n
å
p( m ).
(5)
n =1 mÎS n
The main problem in calculating the characteristics (4) and (5) is to calculate p( m ), m Î S , since it is impossible for this model to find the explicit solution of the corresponding system of global equilibrium equations (SGEE) for stationary state probabilities. This complicates the problem solution for a high-dimension PS (2). Therefore, we propose another approach, based on the fact that QoS parameters (4) and (5) can be easily defined in terms of the probabilities of lumped states S n , n = 0, 1,¼, N , which unite microstates from PS (2) with identical numbers of busy channels. In other words, we can find the necessary characteristics by using the following probabilities of lumped states: (6) p( n ) : = å p( m ), n = 0, 1,... , N . mÎS n
It is obvious that
N
å p( n ) = 1.
(7)
n= 0
607
Thus, we take into account (4)–(7) and execute Pi : =
N
å (1- a i ( n )) p (n ),
(8)
i = 1,... , K ;
n= 0
N
N av : = å np ( n ) .
(9)
n =1
Thus, without determining the stationary distribution of the original model, we can find the QoS parameters (4) and (5) if it is possible to determine p( n ), n = 0, 1,¼, N . To this end, split PS (2): S=
N
U Sn,
n= 0
S n I S n ' = Æ, n ¹ n¢,
(10)
where the sets S n are defined above (see formula (4)), i.e., a class of states S n contains microstates m Î S at which the number of busy channels is n, n = 0, 1,¼, N . Statement 1. If the local balance condition is satisfied in the system, then the stationary probabilities of lumped states are defined as ö 1 n -1 æ K (11) p ( n ) = Õ ç å r j a j ( i ) ÷ p ( 0), n = 1,K , N , ÷ n ! i = 0 ç j =1 è ø -1
æ N 1 n -1 æ K öö where p ( 0) = ç å Õ ç å r j a j ( i ) ÷ ÷ , r i : = l i / m i , i = 1,¼, K . Hereinafter, we assume that ç ç ÷÷ n! øø è n = 0 i = 0 è j =1 b
Õ xi : = 1 if
b
å xi : = 0
and
i=a
a > b.
i=a
To prove the statement, we will need the following lemma. LEMMA. If the local balance condition is satisfied in the system, then K
å r i a i ( n - 1)p ( n - 1) = np ( n ),
n = 1,... , N .
(12)
i =1
Proof. We will use the scheme proposed in [16]. Since the local balance condition is satisfied in the system, taking into account (3) we obtain that the system of local balance equations for the states m Î S n , n = 1,¼, N , has the following form: if i = j, ì m j p( m ) (13) r j a j ( n - 1) p( m - e i ) = í î ( m j + 1) p( m - e i + e j ) if i ¹ j. Summing up both sides of (13) over all the possible i Î {1,¼, K } and m Î S n and taking into account that conditions m and m – e i + e j belong to the same class S n , we find K
å å
i , j =1 mÎS n
K
r j a j ( n - 1) p( m - e i ) = å
å mi p( m ).
(14)
i =1 mÎS n
The left side of (14) is representable as follows: K
å å
i , j =1 mÎS n
K
K
j =1
i =1 mÎS n
r j a j ( n - 1) p( m - e i ) = å r j a j ( n - 1)å
å
p( m - ei )
K
= å r j a j ( n - 1) p ( n - 1) .
(15)
j =1
In the last transformations, in regrouping the terms of the sum, relationship (6) was taken into account, as well as the following fact: for all the states m Î S n , n = 0, 1,K , N , the values of
608
K
å r j a j ( n ) are j =1
identical.
Transform the inner sum in the right-hand side of (14) as follows:
å
mÎS n
mi p( m ) =
å
mÎS n
mi
p( m ) p (n ) . p (n )
(16)
According to the definition of conditional probability, æ P (m | n ) = P ç m ç è
ì p( m ) if m Î S n , ö ï å mi = n ÷÷ = í p ( n ) i =1 ø ï 0 otherwise. î K
(17)
Then (16) and (17) yield K K æ ö æK ö mi p( m ) = å ç å mi P ( m | n ) ÷ p ( n ) = å E ( mi | n )p ( n ) = E ç å mi | n ÷ p ( n ) = n p ( n ) , ç ÷ ç ÷ i =1 mÎS n i =1 i =1 è mÎS n è i =1 ø ø K
å å
(18)
where E( × | × ) denotes conditional mathematical expectation. Hence, taking into account (14), (15), and (18), we conclude that (12) is true. The Proof of Statement 1. Considering relationships (12) and the normalization condition (7), we prove formulas (11). As was mentioned above, a number of classical CAC strategies can be obtained from the proposed strategy in special cases. Indeed, putting in (11) a i ( n ) = 1 for any i = 1,¼, K and n = 0,¼, N - 1 yields the well-known distribution for the number of busy channels in the classical Erlang model: p (n ) =
rn n!
æ N ri ×ç å ç è i=0 i!
K ö ÷ , n = 0, 1,K , N ; r : = å r i . ÷ i =1 ø
(19)
Applying certain algebraic transformations proves that for a multiparameter CAC (see (1)), the stationary distribution of lumped states can be found from K ö p ( 0) K æç p (n ) = r- å r j ÷ Õ ÷ n ! i = m + 1ç j=i è ø
N i - N i+ 1
K æ ç r - å rl ç l=m+1 è
ö ÷ ÷ ø
n- N m
(20)
if N m + 1 < n < N m , m = 1,K , K , where N K + 1 : = 0 and p( 0) can be found from the normalization condition (7). Now assume that the local balance condition is not satisfied in the system. Then the following approach can be employed. Taking into account (3), we obtain that the SGEE for conditions m Î S n -1 has the following form: K æK ç å l i a i ( n - 1) + å mi m i ç i =1 è i =1
K K ö ÷ p( m ) = å l i a i ( n - 2) p( m - e i ) + å ( mi + 1) m i p( m + e i ) . ÷ i =1 i =1 ø
(21)
To simplify the notation, we assume that the states m, m - e i , and m + e i appearing in (21) belong to PS (2); otherwise, the corresponding terms are equated to zero. Summing up both sides of (21) over all the possible m Î S n -1 , collecting similar terms, and taking into account the structure of the SGEE, we obtain K
å l i a i ( n - 1) å i =1
mÎS n- 1
K
p( m ) = å
å
i =1 mÎS n
mi m i p( m ) .
(22)
Rearrange Eq. (22) with regard for (6) as K
K
i =1
i =1 mÎS n
p ( n - 1)å l i a i ( n - 1) = å
å mi m i p( m ).
(23) 609
Equations (12) can be obtained from (23) for m i = m j , i, j = 1,¼, K . Even if the service rates of heterogeneous calls differ insignificantly, the computation procedure described above can be used to find the QoS parameters of the system. And if the service rates of heterogeneous calls are significantly different, various schemes of “unifying” (“averaging”) their values can be used, followed by the proposed procedure. From the practical point of view, the following three general values are of primary interest: (i) m : = max { m1 , K, m K }; (ii) m : = min { m1 ,... , m K }; K 1 K (iii) m = å r i , where L : = å l i . L i =1 i =1 For each scheme of (these and other) “averagings,” the approximations can be analyzed for accuracy numerically since there is no analytical solution. For low-dimensional models, the exact solution can also be found from the SGEE.
2. APPLYING THE MODEL IN COMMUNICATION NETWORKS The model studied here is widely applied in systems of transfer and handling of messages of different types. Below, we will consider two examples and some special cases. First, consider the model of a voice/data wireless network [13]. A short description of the model is as follows. An isolated cell of a multiservice network processes voice calls and data calls. There are four types of calls in it: handover voice calls ( hv-calls), new voice calls ( ov-calls), handover data calls (hd -calls), and new data calls ( od -calls). Each cell has N > 1 radio channels. They are shared by Poisson flows of hv-, ov-, hd-, and od-calls. The rate of x-calls is l x , x Î {hv, ov, hd , od }. The average processing rate for one (new or handover) voice call is m v , and the corresponding parameter for a (new or handover) data call is m d . The following multiparameter CAC strategy is introduced in the system. Three parameters, N od , N hd , and N ov , satisfying the inequality 0 < N od £ N hd £ N ov £ N are determined. The following rules of accepting heterogeneous calls are applied for the proposed strategy: · If at the moment of arrival of an x-call the number of busy channels in the system is no greater than N x , it is accepted for service; otherwise it is refused, x Î{hd , od , ov }; · If there is at least one free channel in the system at the moment of arrival of an hv-call, it is accepted for service; otherwise it is refused. This model was investigated earlier in [13, 17] with the use of various numerical methods. Applying the approach proposed above allows deriving analytical results for the model under study. Indeed, this CAC strategy is a special case of the randomized strategy described above, where K = 4 and parameters a i ( n ) are defined similarly to (1). Then applying algorithm (21) allows proving the following statement. Statement 2. The stationary probabilities of lumped states in the given model for m v = m d are defined as follows: ì p ( 0) ï n! ï ï p ( 0) ï p (n ) = í n ! ï p ( 0) ï n! ï p ( 0) ï î n!
r n if 1 £ n £ N od - 1, r N od ( r - r od ) m - N od if N od £ n £ N hd - 1, m - N hd
if N hd £ n £ N ov - 1,
N ov - N hd
r hv-
r N od ( r - r od ) N hd - N od r v
r N od ( r - r od ) N hd - N od r v
n N ov
(24)
if N ov £ n £ N ,
where r od : = l od / m d , r hd : = l hd / m d , r ov : = l ov / m v , r hv : = l hv / m v , r v : = r ov + r hv , r d : = r od + r hd , and r : = rv + rd . Then the loss probabilities for heterogeneous calls are defined as Px =
610
N
å
n= N x
p( n ), x Î {od , hd , ov, hv }, N hv : = N .
(25)
The average number of busy channels of a cell can be found from (9). As is seen from (9), (24), and (25), the calculation of QoS parameters becomes much simpler as compared with numerical methods [13, 17]. The approximate algorithms proposed in [17] are highly accurate. Note that these algorithms are applicable for cell models where the following conditions are satisfied: l v >> l d , m v >> m d . A special case of the model is the monoservice model analyzed in [14]. Poisson flows of new and handover calls of one class of messages are serviced in it. The rate of x-calls is l x , x Î{o, h }. If there is at least one free channel at the moment of arrival of an h-call, it is accepted and one of the free channels is appointed for its service; otherwise, the h-call is lost. An arrived î-call is accepted with probability a( n ) if the number of busy channels at the moment of arrival of such a call is n, n = 0, 1,¼, N - 1, a( N ) : = 0; with additional probability 1- a( n ), the arrived î-call is blocked. The rate of service of new (handover) calls is m o ( m h ), and, generally speaking, m o ¹ m h . The proposed method yields the following algorithm to calculate the required QoS parameters: Ph = p( N ), P0 = where
(26)
N
å (1- a( n )) p ( n ) ,
(27)
n= 0
-1
æ N 1 i -1 ö 1 n -1 p ( n ) = Õ ( a( i ) r o + r h ) p ( 0 ) , p ( 0 ) = ç å Õ ( a( j ) r o + r h ) ÷ . ç i=0 i! j=0 ÷ n! i=0 è ø
(28)
Formulas (26)–(28) were proposed in [14] based on some heuristic reasons; however, the authors of that study assumed that the algorithm developed above is approximate. We have shown that it is exact. Now, let us consider a model of an integrated network of transmitting narrowband voice calls (v-calls) and broadband data calls (d -calls) [9]. A short description of the model is as follows. The integrated voice and data transmission network contains N > 1 identical and parallel channels. The traffic of narrowband v-calls is described by the Poisson law with mean l v , every newly arrived v-call needs only one channel for service (transfer). The traffic of broadband d -calls is a Poisson flow with mean l d . Everyone newly arrived d -call needs b, 1 < b < N , channels simultaneously, and all the channels simultaneously begin and complete the service of the call. The service time for v-calls (d -calls) has exponential distribution with mean m v ( m d ) . The proposed CAC strategy is defined as follows. If the number of free channels at the moment of arrival of a d -call is no less than b, it is accepted for service; otherwise, the arrived call is blocked with unit probability. If the number of busy channels at the moment of arrival of a v-call is n, it is accepted with probability a( n ), 0 £ a( n ) £ 1, n = 0, 1,¼, N - 1, and is blocked with additional probability; and a( N ) : = 0. Given the specific values of the parameters a( n ), n = 0, 1,¼, N - 1, a number of the well-known CAC strategies can be obtained [9]. Statement 3. The stationary probabilities of lumped states in the considered model for m v = m d are p ( n ) = rn p ( 0), n = 0, 1,... , N ,
-1
(29)
æ N ö where p( 0) = ç å rn ÷ , and the parameters rn , n = 0, 1,¼, N , are defined by the following recurrences: ç ÷ è n= 0 ø ì r nv n -1 × Õ a( j ) if 0 £ n £ b - 1, ï ï rn = í n ! j = 0 ï 1 ( r a( n - 1) r + b r r n -1 d n - b ) if b £ n £ N , ïî n v
(30)
where r x : = l x / m x , x Î {v, d }. Hence, considering (30) we find that the required QoS parameters are defined by the following relations: N
Pv = p ( 0) å (1- a( n )) rn ; n= 0
Pd = rN p( 0);
N
N av = p( 0) å nrn . n =1
Note that the formulas considered in the paper allow performing a reliable analysis (with little computational effort) of the QoS parameters of models of any dimension in any range of load parameters of heterogeneous traffic. 611
CONCLUSIONS In the paper we have proposed an analytical approach to the analysis of a multidimensional Erlang model with a randomized CAC strategy. This approach can be easily adapted to a multirate Erlang model. Unlike the well-known numerical methods, it does not need a large space of states of the model to be generated; hence, the required QoS parameters are found by explicit formulas. A similar model was analyzed in [12] for the special case where all the calls are identical in service time. The results presented in [12] can easily be obtained from the results of the present paper. Applying the results in communication networks substantially simplifies the solution of some problems in their analysis and optimization [13]. It has been proved that the results obtained earlier for wireless communication networks and considered approximate [14] are exact. They allow formulating and solving various optimization problems for a multidimensional Erlang model (such as choosing appropriate values of the parameters of CAC introduced), and support a prescribed quality of service of heterogeneous calls. These problems form the basis for further research.
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