Analysis of priority channel assignment schemes in mobile cellular communication systems: a spectral theory approach

Performance Evaluation 59 (2005) 199–224

Analysis of priority channel assignment schemes in mobile cellular communication systems: a spectral theory approach Vicent Pla∗ , Vicente Casares-Giner Departamento de Comunicaciones, Universidad Polit´ecnica de Valencia (UPV), ETSIT Cam´ı de Vera s/n, 46022 Valencia, Spain Available online 11 September 2004

Abstract A queueing system with two arrival streams is considered in this paper. A number of schemes assigning different priorities to each of the two arrival streams have been modeled. One of the streams (stream 1) is considered to require a higher priority to access the server than the other (stream 2). On the other hand, stream 1 has stringent constraints with regard to the waiting time, and thus it is treated according to a loss model whereas stream 2 is handled following a delay model. The analysis of the system is carried out by applying the generating function method combined with spectral expansion tools. This approach offers better numerical efficiency and stability over a wider range of system loads than the widely used matrix-geometric approach. The model is applied to a mobile cellular communications system where the call setup requests form the stream 2 and handover requests form the stream 1. © 2004 Elsevier B.V. All rights reserved. Keywords: Queuing analysis; Spectral expansion; Priority system; Mobile cellular communications; Channel assignment

1. Introduction In cellular systems, mobile terminals (MTs) are linked to a base station (BS) with a radio channel. The radius of the area covered by a BS ranges from a few meters (microcell) to several kilometers (macrocell). During a conversation, an MT can move across cells and the cellular system should guarantee the continuity of the conversation in progress executing a handoff procedure each time the MT crosses the cell boundaries. During the handoff procedure the MT releases the channel in the originating cell and ∗

Corresponding author. Tel.: +34 963 879733; fax: +34 963 877309. E-mail addresses: [email protected] (V. Pla); [email protected] (V. Casares-Giner).

0166-5316/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.peva.2004.07.001

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it is assigned a new channel in the destination cell. If the latter is not possible due to unavailability of channels, the call will be forcedly terminated. In contrast, for the new calls, unavailability of channels imposes no call establishment or delayed call establishment. As far as the user is concerned, no call set up or delayed set up is preferable to the disruption of an ongoing call. Consequently some kind of priority for handoff requests is required. While the subject of channel assignment in the context of cellular systems and its analytical treatment is not a new topic — first studies date back to the late 1980s and early 1990s [1–6]— a great deal of references in this field can still be found in the literature (see for instance [7–9], to name a few). This is probably due to the enormous growth experienced by the demand of personal telecommunication services in recent years. Since radio spectrum is a scarce resource, the higher capacity required is mainly obtained by means of cell size reduction. In addition to the infrastructure costs, the price to pay for cell size reduction is a higher handoff rate and more frequent handoffs per call. Consequently, it is increasingly important to find clever methods to assign channels to be able to keep a low probability of forced termination while making an efficient use of resources. On the other hand, more sophisticated schemes involve more complex models requiring efficient mathematical tools for their analysis and dimensioning. The forthcoming 3G mobile cellular communications systems will have the property of soft-capacity or graceful degradation. While this could allow more flexible methods of call admission schemes, it does not preclude the existence of a limit in the number of active terminals per cell [10,11]. Moreover, the advent of 3G systems will give rise to the emergence of new services and, as different services will have different needs and traffic characteristics [12], richer schemes for channel assignment will be required. Thus, developing mathematical tools suitable for the analysis of such schemes, constitutes an important research topic. Many protocols have been designed to limit the probability of forced termination of an ongoing call at the expense of lowering performance for new calls. Most of these protocols are based on the Reserve Guard Channel concept, i.e. a fixed number of channels in each cell are reserved exclusively for handoff calls [1]. An enhancement of this algorithm allows a waiting buffer for new calls which are not admitted into service immediately upon arrival. This algorithm has been analyzed in [2], and in later papers such as [13] and [14]. The dynamics of this algorithm can be described as follows. The base station handles two types of calls: calls that are transferred from another base station (class 1 or handoff calls) and calls originated within the cell (class 2 or fresh calls). Assuming a total of m + n channels per cell, calls originating within the cell are queued if the number of channels available is not greater than n; n being the number of guard channels. Handoff calls are always admitted provided there is least one idle channel. Gu´erin [2] models the system using a quasi-birth-death process (QBD) and, using a rather complicated procedure, he obtains analytical expressions for the steady state probabilities, which in turn gives the blocking probability and mean waiting time for class 1 and 2, respectively. In another work, Daigle and Jain [13] build a model by observing that the queueing system for the originating calls can be analyzed as an M/G/1 queueing system with exceptional first service and phase-type service time distributions. They also observe that the infinitesimal generator of the underlying QBD process is of G/M/1 type [15]. Combining these two observations, they construct three different methods for computing parameters of interest. The study performed by Casares and Holtzman [16] and Casares-Giner [17] in the context of land mobile trunking systems, deals with system models which are quite close to those in this paper. By means of a matrix-geometric approach [15], in [16] the authors obtain the mean waiting time and the blocking probabilities for dispatch (low priority) and interconnect (high priority) calls, respectively. In [17], the waiting time distribution under FIFO discipline for dispatch calls is obtained. The models in both references, [16] and [17], consider different exponential service rates for dispatch and interconnect calls.

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Virtually all these models fall within the category of QBD processes. The most widely used approach to solve such models is perhaps the matrix-geometric approach. This approach relies on determining the minimal positive solution, R, of a nonlinear matrix equation; the equilibrium is then expressed in terms of powers of R. In [18] the author claims that under certain conditions, the iterative procedure used to find R poses some problems in terms of numerical precision and computational cost. The author also proposes and alternative method, namely the espectral expansion approach, that overcomes the aforementioned limitations. The spectral expansion approach is based on expressing the equilibrium distribution of the process in terms of the eigenvalues and left eigenvectors of a certain matrix polynomial [18]. A quite similar approach is that followed by Keilson and Ibe [14], who propose a novel and concise method of analysis by employing the generating function method in conjunction with matrix spectral tools for the underlying QBD process. In this paper, we describe and analyze three new algorithms for channel assignment with handoff priority. The analysis is carried out using the approach proposed in [14] and obtaining useful expressions for the blocking probability for class 1 traffic, mean waiting time for class 2 traffic, as well as other parameters of interest. Lastly, a numerical analysis is performed to study the advantages of the approach employed for the analysis and to determine under which conditions this advantage is significant compared to the widely used matrix-geometric approach. The remaining of the paper is organized as follows. In Section 2, a description of the new three algorithms is presented. The analysis is carried out in Sections 3 and 4. Section 5 reports some numerical results and compares the spectral expansion and the matrix-geometric methods in terms of numerical efficiency and precision. A summary and concluding remarks are given in Section 6.

2. System model This section details the way in which the channel assignment algorithms analyzed in this paper work. The modeling assumptions are subsequently described and state transition diagrams for each algorithm are given. The system model considers an isolated cell, which is not a limitation since, from an analytical viewpoint, extension to a network of cells in statistical equilibrium is straightforward (see [19] for instance). Two types of calls arrive to the target cell: handoff and fresh calls. We consider a fixed total number of channels per cell m + n. When a request arrives, the channel assignment algorithm decides whether to assign it a channel or not. The decision is based on the type of call and the occupancy state of the pool of channels. New calls are queued in case no channel is assigned at the arrival instant, whereas handoff calls are treated according to a lost call model. 2.1. Channel assignment algorithms For the operation of the channel assignment schemes considered in this paper, channels are logically split into two groups and denoted as primary and secondary groups. Such partition is not physical, i.e. every physical channel may belong to either the primary or the secondary group at different times. This can be seen as every channel having a flag that indicates whether it is on the primary or secondary group at any given time. Following this representation, a busy channel is switched from one group to the other by simply toggling the flag associated to the channel. A pictorial example is given in Fig. 1 that depicts a

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Fig. 1. Sample path of channel assignment in HRMA; m = 4, n = 2.

sample path of the channel assignment scheme in the HRMA algorithm (see the algorithm specification below). In the example, there are a total of 6 channels and shaded boxes stand for an idle channel. Note that the queue of new calls is not shown in Fig. 1. In the sequel we use the following notation: • • • • •

m, the maximum number of busy channels in the primary group; n, the maximum number of busy channels in the secondary group; p, the number of busy channels in the primary group (0 ≤ p ≤ m); s, the number of busy channels in the secondary group (0 ≤ s ≤ n); q, the number of new call requests in the queue.

2.1.1. Handoff reserve margin algorithm (HRMA) This is the algorithm already described and analyzed in [2,13,14]. Alternatively, it can be described as follows. Handoff calls are offered in a sequential search, first to the primary group, and then (if p = m) to the secondary group. If both groups are full (p = m and s = n), the handoff call is lost. Whenever a primary channel is released, a call in progress in the secondary group (chosen at random) is switched to the primary. New calls can only be served by a primary group channel. If there are no idle channels in this group the request is queued. A flowchart of this algorithm is shown in Fig. 2. 2.1.2. Handoff calls overflow from primary to secondary with rearrangement (HOPSWR) Handoff and new calls are offered in the same way as in the HRMA. However, if there are some new call requests in the waiting line, no reassignment from the secondary to the primary group is done. A flowchart of this algorithm is shown in Fig. 3. 2.1.3. Handoff calls overflow from primary to secondary (HOPS) Handoff and new calls are offered in the same way as in the two previous algorithms, and in contrast, no type of rearrangement is performed. A flowchart of this algorithm is shown in Fig. 4.

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Fig. 2. HRMA flow chart.

2.1.4. Handoff calls overflow from secondary to primary (HOSP) As in the previous algorithms, new calls are only offered to the primary group. Handoff calls are offered first to the secondary group and then (if s = n) to the primary group. If both groups are full, the handoff call is lost. No rearrangement is performed. A flowchart of this algorithm is shown in Fig. 5.

Fig. 3. HOPSWR flow chart.

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Fig. 4. HOPS flow chart.

2.2. Mathematical model For the sake of tractability, we make the common assumptions of Poisson arrivals and negative exponential distribution for service time. We denote by λ1 and λ2 the arrival rate of handoff and new calls, respectively. The aggregated arrival rate is denoted by λ = λ1 + λ2 . The mean service time for both type of calls is µ−1 . Under these assumptions the system can be modeled by a bidimendional Markov process.

Fig. 5. HOSP flow chart.

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Fig. 6. HRMA state transition diagram. Primary group, m = 4; secondary group, n = 2.

2.2.1. HOPSWR Let (r, k) denote a state of the system; r = p + s is the total number of busy channels and k = q is the number of new call requests in the queue. Fig. 7 shows an example of the state transition diagram when m = 4 and n = 2. Although the analysis of the HRMA algorithm is not carried out in this paper, an example of the state transition diagram is provided in for comparison purposes with the proposed algorithms (Fig. 6). 2.2.2. HOPS Let (r, k) denote a state of the system; r = p + q is the total number of new calls in the system (in service plus in queue) and k = s is the number handoff of calls in the system (in service). Fig. 8 shows an example of the state transition diagram when m = 4 and n = 2. 2.2.3. HOSP We use the same state representation (r, k) as in the previous algorithm. Fig. 9 shows an example of the state transition diagram when m = 4 and n = 2.

Fig. 7. HOPSWR state transition diagram. Primary group, m = 4; secondary group, n = 2.

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Fig. 8. HOPS state transition diagram. Primary group, m = 4; secondary group, n = 2.

Fig. 9. HOSP state transition diagram. Primary group, m = 4; secondary group, n = 2.

3. Analysis In this section, the model for HOPS algorithm is analyzed in detail. The analysis of HOPSWR and HOSP is described in the next section in a more concise manner. The analysis employs the generating function method in conjunction with matrix spectral tools for the underlying QBD process. Let prk denote the steady-state probability of the process being in state (r, k), let P Tr = [pr0 , pr1 , . . . , prn ] and let the infinitesimal generator of the Markov process resulting from considering states in P Tr be Q (r ≥ m) and QB (r = 0, . . . , m − 1): 

∗ λ1 0 µ ∗ λ  1   0 2µ ∗  Q=. . .  .. .. ..   0 0 0 0 0 0

 0 0 0 0   0 0  , .. ..  . .    · · · · · · ∗ λ1  · · · · · · nµ ∗ ······ ······ ······ .. .

 ∗ 0 0 ······ 0 0 µ ∗ 0 ······ 0 0      0 2µ ∗ · · · · · · 0 0    QB =  . . . .  . . .. ..  ..  .. .. ..     0 0 0 · · · · · · ∗ 0   0 0 0 · · · · · · nµ ∗ 

where the asterisks (*) denote the required values so that the matrix rows sum to 0.

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The global balance equations can be written as −λP T0 + P T0 QB + µP T1 = 0T ,

(1)

λP Tr−1 − (λ + rµ)P Tr + P Tr QB + (r + 1)µP Tr+1 = 0T ,

0 < r < m,

λP Tm−1 − (λ2 + mµ)P Tm + P Tm Q + mµP Tm+1 = 0T , λ2 P Tr−1 − (λ2 + mµ)P Tr + P Tr Q + mµP Tr+1 = 0T ,

(2) (3)

m < r,

(4)

where 0 is a column vector of zeros.  Now we introduce the Z-domain for convenience. Let P T (z) = r≥m P Tr zr−m , from Eqs. (3) and (4) it follows that P T (z)[(λ2 (1 − z) + mµ(1 − z−1 ))I − Q] = λP Tm−1 − mµz−1 P Tm , where I stands for the identity matrix of appropriate dimension. Thus P T (z) = lim [λP Tm−1 − mµw−1 P Tm ]M(w)−1 , w→z

(5)

where M(z) := (λ2 (1 − z) + mµ(1 − z−1 ))I − Q. In (5) the limit notation is introduced since there are some values in |z| ≤ 1 where the matrix M(z) is singular and M(z)−1 is not defined. By introducing the spectral representation of M(z) and having in mind that [20] f (Q) =

n+1 

f (γj )J j ,

j=1

we obtain



P T (z) = lim (λP Tm−1 − mµw−1 P Tm ) · w→z

n+1  j=1

 Jj , λ2 (1 − w) + mµ(1 − w−1 ) − γj

(6)

where γj are the eigenvalues of Q, and J j are defined by Jj =

uj vTj vTj uj

,

uj and vTj being the right and left eigenvectors of Q, respectively. The proof of the following theorem is given in Appendix A. Theorem 1. The eigenvalues (γj , j = 1, . . . , n + 1) of matrix Q are all real, distinct and non-positive.

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In the sequel eigenvalues γj will be considered to be numbered in descending order, i.e. 0 = γ1 > γ2 > · · · > γn+1 . Besides, since Q is an infinitesimal generator, we have that u1 = 1 and v1 = θ, where 1 is a column vector whose elements are all 1 and θ T = [θ0 , . . . , θn ] denotes a vector whose elements are the steadystate probabilities of the Markov process associated with Q, i.e. n

−1 (λ1 /µ)k  (λ1 /µ)l θk = , k! l! l=0

k = 0, . . . , n

and then J 1 = 1θ T . 3.1. Complementary equations As it can be observed in Eq. (6), P(z) is not completely specified since it depends on P m−1 and P m . Furthermore, probability vectors {P r , r < m} are not comprehended in P(z) and thus need to be obtained separately. On the whole, P 0 through P m are unknown vector probabilities making a total amount of (m + 1)(n + 1) unknown scalar probabilities. Those probabilities are obtained using the following conditions: Boundary equations: m(n + 1) equations. The m(n + 1) scalar equations stem from the m vectorial equations in (1) and (2) Singularities of M(z): n + 1 equations. A necessary condition for the existence of the limit in (6) is that every root z (|z| ≤ 1) of equation λ2 (1 − z) + mµ(1 − z−1 ) − γj = 0,

j = 1, . . . , n + 1

(7)

be also a root of (λP Tm−1 − mµz−1 P Tm )uj = 0T ,

j = 1, . . . , n + 1.

(8)

It is easy to check that the above is also a sufficient condition. The following proposition states that this condition will give a total n + 1 equations. Proposition 2. For each eigenvalue of Q (γj j = 1, . . . , n + 1) Eq. (7) has exactly one zero in the region |z| ≤ 1 which is given by the expression 

1 2 zj = mµ + λ2 − γj − (mµ + λ2 − γj ) − 4mµλ2 . (9) 2λ2 A proof of this proposition is given in Appendix B. Combining the above proposition and (8) we conclude that the following n + 1 equalities are a necessary and sufficient condition for the existence of the limit in (6): T T (λP Tm−1 − mµz−1 j P m )uj = 0 ,

j = 1, . . . , n + 1,

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where zj ’s are defined by (9). When j = 1 (γ1 = 0, u1 = 1) the above equation becomes (λP Tm−1 − mµP Tm )1 = 0T .

(10)

Eq. (10) can also be obtained by adding all equations in (1) and (2) and then right multiplying by 1. Thus, only n (j = 2, . . . , n + 1) new equations can be obtained from the singularities of M(z) Normalization equation: 1 equation. Lemma 3. Let Pr (z) := P T (z)1; then Pr (z) =

mµ P T 1. mµ − λ2 z m

Proof. From (6) and the fact that J j · 1 = 0T if j = 1 (see (C.1) in Appendix C) it follows that  P T (z)1 = lim (λP Tm−1 − mµw−1 P Tm ) · w→z

 J 11 ; λ2 (1 − w) + mµ(1 − w−1 ) − γ1

noting that γ1 = 0, J 1 1 = 1θ T 1 = 1 and using (10) we obtain  Pr (z) = lim

w→z

 mµP Tm 1(1 − w−1 ) mµ mµ = lim PT 1 = P Tm 1. w→z mµ + λ2 1−w−1 m λ2 (1 − w) + mµ(1 − w−1 ) mµ − λ z 2 1−w


Now we write the normalization equation as 1=

n+1 

pji =

j≥0 i=1

 j≥0

P Tj 1

=

m−1  j=0

P Tj 1

+

 j≥m

P Tj 1

=

m−1 

P Tj 1 + Pr (1)

j=0

and the application of Lemma 3 enables us to write 1=

m−1 

P Tj 1 +

j=0

mµ P T 1. mµ − λ2 m

(11)

The m(n + 1) boundary equations plus the n equations obtained from the singularities of M(z) plus the normalization equation provide a total number of (n + 1)(m + 1) equations. 3.2. Parameters of interest Proposition 4. The mean waiting time (W2 ) for low priority customers is given by W2 =

mµ P T 1. (mµ − λ2 )2 m

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Proof. We first obtain the mean number of type 2 customers in the waiting line (Q2 ) and then Little’s Law [21] is applied: d mµλ2 Pr (z) = P T 1. z→1 dz (mµ − λ2 )2 m

Q2 = lim Thus W2 =

Q2 mµ = P T 1. λ2 (mµ − λ2 )2 m

Note that for Q2 and W2 are finite if λ2 < mµ, which is the stability condition for the system. Let us now establish some results that will be used subsequently.




Proposition 5. n+1 

γj−1 J j = 1θ T − (1θ T − Q)−1 .

j=2

This proposition is proven in Appendix C. Lemma 6. P T (1) =

mµ P Tm 1θ T + (λP Tm−1 − mµP Tm )(1θ T − Q)−1 . mµ − λ2

Proof. With simple algebraic manipulation of (6) and recalling that γ1 = 0, J 1 = 1θ T it follows that n+1  mµ T T T T P (1) = P 1θ − (λP m−1 − mµP m ) γj−1 J j ; mµ − λ2 m j=2 T

applying Proposition 5 and using Eq. (10) we can rewrite the above equality as P T (1) =

mµ P T 1θ T + (λP Tm−1 − mµP Tm )(1θ T − Q)−1 . mµ − λ2 m




In the following propositions we obtain expressions for the parameters of interest. Proposition 7. The probability that m + k (0 ≤ k ≤ n) servers are busy when a low priority call is queued, is given by P(m + k) =

mµ P T 1θk + (λP Tm−1 − mµP Tm )(1θ T − Q)−1 ek+1 , mµ − λ2 m

where ei (i = 1, 2, . . .) denotes the column vector with all its entries 0 except the ith one which is 1.

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Proof. Since P(m + k) = P T (1)ek+1 , applying Lemma 6 and introducing θk = θ T ek+1 establishes the formula.
Proposition 8. The blocking probability (Pb ) for high priority (type 1) customers is given by mµ P T 1θn + (λP Tm−1 − mµP Tm )(1θ T − Q)−1 en+1 . Pb = P(m + n) = mµ − λ2 m

(12)

Proof. We first note that Pb = P T (1)en+1 and then the proof proceeds in much the same way as in the previous proposition.
Remark 9. The term (1θ T − Q)−1 ek in (12) can be computed as the solution x to the linear system (1θ T − Q)x = ek , which has a lower computational cost than inverting a matrix. Proposition 10. The mean sojourn time in congestion state for low priority traffic (type 2 arrivals can not be served immediately upon arrival) is given by E[T2c ] =

mµ P Tm 1 . λ(mµ − λ2 ) P Tm−1 1

Proof. The system is in congestion state for low priority traffic when the process is in the states {(r, k) : r ≥ m}. Therefore, in the closed interval [0, T ] the time spent in congestion is Pr (1)T + O(T ), and the number of times the system enters into congestion is λP Tm−1 1T + O(T ). Hence, the mean time spent in each congestion period is Pr (1)T + O(T ) λP Tm−1 1T + O(T ) Letting T → ∞ and substituting Pr (1) by the expression given in Lemma 3 we finally obtain the desired result.


4. Other algorithms 4.1. HOPSWR Some changes in notation are required. Let P Tk = [pm,k , pm+1,k , . . . , pm+n,k ] (k ≥ 0),  0 0 0 ... 0 0  1 −1 0 . . . 0 0       0 1 −1 . . . 0 0    D=. . . .  . . . . . . . . . . . . . .      0 0 0 . . . −1 0   0 0 0 . . . 1 −1 

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and QB = Q + mµD. The global balance equations can be written as −λ2 P T0 + P T0 QB + mµP T1 = 0T , λ2 P Tk−1 − (mµ + λ2 )P Tk + P Tk Q + mµP Tk+1 = 0T ,

k > 0,

where we have used the fact that λpm−1,0 = mµpm,0 . Then P T (z) =

 k≥0

 P Tk zk = lim mµP T0 [D + (1 − w−1 )I] · w→z

n+1  j=1

Jj  . (13) λ2 (1 − w) + mµ(1 − w−1 ) − γj

In an analogous manner to the proof of Lemmas 3 and 6, we obtain Pr (z) =

P T (1) =

mµ P T 1, mµ − λ2 0 mµ P T 1θ T + mµP T0 D(1θ T − Q)−1 . mµ − λ2 0

4.1.1. Complementary equations From the boundary equations it easily follows that pr,0

m! = r!

 r−m λ P T0 e1 , µ

r = 0, . . . , m − 1.

The normalization equation, which is written as P T0



m−1  m!  λ r−m mµ P T0 e1 e1 = 1, 1+ λ2 − mµ r! µ r=0

and the existence of the limit in (13) when |z| ≤ 1 provides the rest of necessary equations   P T0 D + (1 − zj )I uj = 0, where zj ’s are defined in (9).

j = 2, . . . , n + 1,



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4.1.2. Parameters of interest If we now proceed analogously to the proof of Propositions 4, 7, 8 and 10 the following results are obtained P(m + k) =

mµ P T 1θk + mµP T0 D(1θ T − Q)−1 ek+1 , mµ − λ2 m

mµ P T 1θn + mµP T0 D(1θ T − Q)−1 en+1 , mµ − λ2 m mµ P T 1, W2 = (mµ − λ2 )2 0 1 P T0 1 E[T2c ] = . λ(mµ − λ2 ) P T0 e1 Pb =

4.2. HOSP The global balance equations for the homogenous levels (r > m) and boundary level (m) are, respectively λ1 (P Tm−1 en+1 )eTn+1 + λ2 P Tm−1 − (λ2 + mµ)P Tr + P Tm Q + mµP Tm+1 = 0T , λ2 P Tr−1 − (λ2 + mµ)P Tr + P Tr Q + mµP Tr+1 = 0T . then

 P T (z) = lim (λ1 (P Tm−1 en+1 )eTn+1 + λ2 P Tm−1 − mµw−1 P Tm ) · w→z

n+1  j=1

Jj . λ2 (1−w)+mµ(1−w−1 )−γj (14)

4.2.1. Complementary equations The boundary equations can be written as λ1 P Tm−1 en+1 + λ2 P Tr−1 1 − rµP Tr 1 = 0T ,



r = 1, 2, . . . , m.

The normalization equation is the same as that of HOPS algorithm, Eq. (11). As before, the existence of the limit in (14), when |z| ≤ 1, provides the remaining necessary equations (λ1 (P Tm−1 en+1 )eTn+1 + λ2 P Tm−1 − mµw−1 P Tm ) · uj = 0,

j = 2, . . . , n + 1,

where zj is defined in (9). 4.2.2. Parameters of interest In an analogous manner to the proof of Lemmas 3 and 6 we obtain P T (1) =

mµ P T 1θ T + (λ1 (P Tm−1 en+1 )eTn+1 + λ2 P Tm−1 − mµP Tm )(1θ T − Q)−1 . mµ − λ2 m

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Fig. 10. HOPSWR: relative error of QoS parameters.

Fig. 11. HOPS: relative error of QoS parameters.

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Fig. 12. HOSP: relative error of QoS parameters.

Using the preceding equality and proceeding in much the same way as in Propositions 7, 8, 4 and 10 we obtain P(m + k) =

mµ P T 1θk (λ1 (P Tm−1 en+1 )eTn+1 + λ2 P Tm−1 − mµP Tm )(1θ T − Q)−1 ek+1 , mµ − λ2 m

Pb =

mµ P T 1θn · (λ1 (P Tm−1 en+1 )eTn+1 + λ2 P Tm−1 − mµP Tm )(1θ T − Q)−1 en+1 , mµ − λ2 m

W2 =

mµ P T 1, (mµ − λ2 )2 m

E[T2c ] =

P Tm 1 mµ . mµ − λ2 P Tm−1 (λ1 en+1 + λ2 1)

5. Numerical evaluation In this section, the spectral expansion and the matrix-geometric methods are compared in terms of numerical efficiency and precision.

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Fig. 13. HOPSWR: computational cost.

5.1. Precision As it is reported in [18], we observed that the matrix-geometric method exhibits precision problems when the system approaches instability. Figs. 10–12 show the relative error of the QoS parameters Pb and W computed by the matrix-geometric method. The values computed with the spectral method were taken as exact. This assumption is validated by comparing in a heavy load condition the numerical value of W2 with the mean waiting time in the queue M/M/m/∞ (recall that m is the maximum number of channels in the primary group). In this comparison, the relative error of the spectral method was low enough (less than 10−3 ) to consider it as exact for our purposes whereas the relative error of the matrix-geometric method was significant. While in general this is an important advantage for the numerical evaluation of priority systems, in the particular case of channel assignment schemes in cellular communications it has a limited practical relevance, since for the load values at which the precision issue arises, the QoS of the system (measured by Pb and W) is below any reasonable minimum. 5.2. Efficiency Here we asses the computational performance of the spectral approach compared to the matrixgeometric method. Computational cost is measured as the number of required floating point operations. The comparison is carried out for the different algorithms under analysis and for a varying size of the system (number of available channels).

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Fig. 14. HOPS: computational cost.

Fig. 15. HOSP: computational cost.

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Fig. 16. HOPSWR: relative computational cost.

In both approaches, matrix-geometric and spectral, numerical computation of state probabilities consist of two parts: 1. States in the non-homogenous levels. These state probabilities are obtained from the boundary equations (see Section 3.1). 2. States in the homogenous levels. Here is where the two approaches differ: the matrix-geometric method relies on determining the rate matrix R, which is a solution of a non-linear matrix equation, and then probabilities are expressed in terms of powers of R; in the spectral method, the Z-transform of probabilities, P T (z), is expressed in terms of the eigenvalues and the eigenvectors of the matrix Q, Eq. (6). In Table 1, the size of the numerical problem to be solved in each of the two parts is shown. Recall that m and n denote the number of channels in the primary and secondary groups, respectively. Note that the size

Table 1 Problem size for each part Algorithm

Part 1

Part 2

HRMA, HOPSWR HOPS, HOSP

O(m + 2n) O(m · n)

O(n) O(n)

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219

Fig. 17. HOPS: relative computational cost.

of the problem is the same for both approaches and the difference between the two approaches lies on how efficiently part 2 of the problem is solved. In Figs. 13–15 the computational cost is plotted as a function of the total number of channels (C = m + n). In contrast, Figs. 16–18 plot the relative computational cost of the matrix-geometric method with respect to the spectral method. From the plots and from the results in Table 1, the following conclusions can be drawn: the numerical problem in part 1 is less costly than that of part 2 (solution of a linear system versus an eigenvalue problem or a non-linear matrix equation). Hence, when the size of both parts is of the same order of magnitude (low C) the relative cost of part 2 is more significant, resulting in a better performance for the spectral approach. However, in algorithms HOPS and HOSP the size of part 1 grows much faster (quadratic versus linear growth) than the size of part 2, as well as faster than its counter part in algorithm HOPSWR. Then, for those algorithms (HOPS and HOSP), when C increases the relative cost of part 1, it outweighs the computational cost of part 2 becoming the dominant part. As a consequence, for high values of the system size C the difference between the two approaches vanishes, and the absolute cost for algorithms HOPS and HOSP is far higher than that of HOPSWR. 6. Conclusions A number of priority channel assignment schemes for new and handoff calls in a cellular environment have been analyzed applying a new method based on spectral theory. From the analysis of the models we obtained useful and numerically efficient expressions for performance evaluation of the schemes. The

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Fig. 18. HOSP: relative computational cost.

priority schemes presented and analyzed in the paper can be applied not only to the channel assignment problem in cellular networks but to many other telecommunication and computer systems that could fit into a two priorities model like the one analyzed here. The analysis approach employed,which is based on spectral tools, has been compared with the matrixgeometric method in terms of precision and numerical efficiency. The numerical analysis revealed that for high system load, the matrix-geometric method is outperformed by the spectral approach, which is shown to be more precise. As a general rule, the spectral method also shows a better performance in terms of numerical efficiency. The latter advantage, however, tends to vanish when the system size grows in algorithms HOPS and HOSP.

Acknowledgements The authors want to express their gratitude to Vicente Hern´andez for his collaboration in Appendix A. This work has been supported by the Spanish Ministry of Science and Technology under projects TIC20010956-C04-04 and TIC2003-08272.

Appendix A Theorem 1. The eigenvalues (γj , j = 1, . . . , n + 1) of the matrix Q are all real, distinct and non-positive.

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221

Proof. Let  

D = diag 1, 



λ1 , µ



λ21 ,..., 2µ2



 λn1  . n!µn 

˜ = DQD−1 is symmetric, and therefore with real eigenvalues. Also, It is straightforward to check that Q ˜ is tridiagonal with all its sub-diagonal elements different from zero and hence its eigenvalues the matrix Q ˜ in addition to real are distinct (see [22, Section 5.377]). By the well-known result that says that Q and Q have the same eigenvalues, we conclude that the eigenvalues of Q are real and distinct. Now it remains to prove eigenvalues are non-positive. According to Gershgorin’s Theorem [22, Section 2.13] the eigenvalues of Q lie within the region

:=

 

λ : |λ − qii | ≤



n+1  j=1j =i

 

|qij |, i = 1, 2, . . . , n + 1 ; 

using the entries of Q we write

:= {λ : |λ − qii | ≤ −qii , i = 1, 2, . . . , n + 1}. Finally, by noting that qii < 0 it is easily seen that is contained in the semi-plane Re z ≤ 0.




Appendix B Proposition 2. For each eigenvalue of Q (γj region |z| ≤ 1, this root is given by

j = 1, . . . , n + 1) Eq. (7) has exactly one zero in the



1 2 mµ + λ2 − γj − (mµ + λ2 − γj ) − 4mµλ2 . zj = 2λ2 Proof. The proof is divided into two steps. We first prove that when j = 1, Eq. (7) has one zero on the unit circle and one outside the unit circle. It is immediate that when j = 1 the zeros of (7) are z1 = 1 and z2 = mµ/λ2 . Obviously, |z1 | = 1, and assuming the system is stable, |z2 | = mµ/λ2 > 1. Secondly, we show that for j = 2, . . . , n + 1 Eq. (7) has exactly one zero in |z| < 1. We now rewrite Eq. (7) as  z − 2

γj mµ mµ z+ +1− = f (z) + g(z) = 0, λ2 λ2 λ2

j = 2, . . . , n + 1,

where f (z) and g(z) are defined as

γj mµ z +1− f (z) = − λ2 λ2 

and

g(z) = z2 +

mµ ; λ2

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since γi is non-positive (Theorem 1), we have |f (z)||z|=1

   2 mµ  γj mµ mµ  = +1− > + 1 ≥ z + = |g(z)||z|=1 . λ2 λ2 λ2 λ2 |z|=1

Hence, applying Rouche’s Theorem [23] we conclude that f (z) and f (z) + g(z) have the same number of roots inside the unit circle. It is evident that f (z) has exactly one root (z = 0) inside the unit circle. Thus, f (z) + g(z) has exactly one root inside the unit circle. Finally, being f (z) + g(z) a second degree polynomial, obtaining the claimed expression for the root is a simpler matter.
Appendix C Proposition 3. n+1 

γj−1 J j = 1θ T − (1θ T − Q)−1 .

j=2

Proof. Let us first show that if i = j vTi uj = 0.

(C.1)

By definition Quj = γj uj and multiplying both sides by vTi it follows that vTi Quj = γj vTi uj , γi vTi uj = γj vTi uj , (γi − γj )vTi uj = 0, and as γi = γj (Theorem 1) we obtain that vTi uj = 0. Using the fact that γ1 = 0 we can write Q=

n+1 

γj J j =

j=1

n+1 

γj J j

j=2

and J1 − Q = J1 −

n+1 

γj J j .

j=2

It is easy to check that (J 1 − Q)1 = 1;

θ T (J 1 − Q) = θ T

(C.2)

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and for j = 2, . . . , n + 1 we use (C.1) to obtain (J 1 − Q)uj = −γj uj ;

vTj (J 1 − Q) = −γj vTj .

Therefore, {1, γ2 , . . . , γn+1 } are the eigenvalues of J 1 − Q, and its right (left) associated eigenvectors are {1, u2 , . . . , un+1 } ({θ T , vT2 , . . . , vTn+1 }). Thus, the right hand term of (C.2) is the spectral expansion of J 1 − Q. Consequently (J 1 − Q)−1 = J 1 −

n+1 

γj−1 J j

j=2

and taking J 1 = 1θ T establishes the formula.




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