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An efficient heuristic for the expansion problem of cellular wireless networks
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Computers & Operations Research 31 (2004) 1769 – 1791
www.elsevier.com/locate/dsw
An e$cient heuristic for the expansion problem of cellular wireless networks Steven Chamberland∗ CRT and Computer Engineering Department, Ecole Polytechnique de Montreal, C.P. 6079, Succ. Centre-Ville, Montreal (Quebec), Canada H3C 3A7
Abstract In this paper we propose a model for the expansion problem of the network subsystem (NSS) of a universal mobile telecommunication system (UMTS) wireless cellular network considering an update in the base station subsystem (BSS). The objective is to minimize the expansion cost of the network subsystem while considering network performance (e.g., call and handover blocking). Since the network expansion problem is a generalization of the design problem, the proposed model can also be used for designing networks. In order to 6nd good solutions, we propose a heuristic based on the tabu search principle. Finally, we present a performance analysis of the proposed heuristic. The analysis shows that quasi-optimal solutions are found with the proposed heuristic. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Cellular networks; Topological expansion; Facility location and type selection; Tabu search
1. Introduction Cellular wireless network operators dedicate an important proportion of their budget to acquire, install and maintain the facilities that carry tra$c from cell sites to switches and gateways. These facilities are often leased from local exchange carriers. The pressure to reduce costs adds new urgency to the search for optimized networks, which can minimize the cost of required facilities while satisfying a set of predetermine constraints. Considering the potential augmentation of the number of customers for the services o
Corresponding author. Tel.: +1-514-340-4711 x 2996; fax: +1-514-340-3240. E-mail address: [email protected] (S. Chamberland).
0305-0548/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0305-0548(03)00119-9
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In a typical UMTS network, the area of coverage is geographically divided into cells and the network topology is hierarchically organized in order to reduce costs. Each cell is implemented with a base transceiver station (node B) that contains the radio transceivers providing the radio interface with mobile stations. One or more node Bs are connected to a radio network controller (RNC) that provides a number of functions related to resource and mobility management as well as operation and maintenance for the overall radio network. One or more RNCs are connected to a mobile switching center (MSC) or switch that control call setup, call routing, while performing many other functions provided by a conventional communications switch. Each MSC provides connectivity to the public switched telephone network (PSTN) for the voice services. Moreover, one or more RNCs are connected to a SGSN (serving GPRS (general packet radio service) support node) or gateway. Each SGSN provides connectivity to an Internet protocol (IP) data network. Typically, the expansion of a cellular wireless networks requires: 1. the analysis of the network performance based on, among others, call and handover blocking; 2. the analysis of radio-wave propagation and/or the 6eld topology to identify a set of new cells to improve the network performance; 3. the expansion of the node Bs to RNCs network while taking into account a certain number of constraints including capacity constraints; 4. the location of new RNCs, MSCs and SGSNs and the selection of their types; 5. the update of the 6xed network connectivity considering topological constraints and the network performance. In this paper, we are interested in the expansion problem of the network subsystem (NSS) of an UMTS wireless cellular network considering an update in the base station subsystem (BSS). The proposed model deals with • locating new RNCs, MSCs and SGSNs; • selecting their types; • updating the network connectivity; while considering the call and handover blocking. Furthermore, we suppose that several classes of service (CoS) are available in the network (e.g., one class for voice services and others for data services). The objective is to minimize the cost of the expansion process, including the cost of the equipments and the installation costs. The literature of Engineering and Operations Research contains many articles relating to NSS planning problems. Some NSS planning studies focus only on a portion of the overall problem, such as, the location problem of controllers and switches [2–4], the assignment problem of cell to switches [5–9] and the topological design and link dimensioning problem [10–12]. Other studies concern these problems with additional features such as diversity or survivability constraints [10,13] and multi-period demand [14]. However, the overall NSS planning problem tackle in this paper was not considered so far. The rest of this paper is organized as follows. In Section 2 we present a model for the expansion problem of the NSS of an UMTS wireless cellular network. In Section 3, we present a heuristic based on the tabu search principle and, in Section 4, numerical results are presented. Conclusion remarks are presented in Section 5.
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Node B
MSC RNC Public switched telephone network (PSTN)
Node B
Node B
Internet protocol (IP) network
Node B
RNC
SGSN
Node B
Fig. 1. UMTS network architecture
2. Problem formulation In this section, we present the model for the expansion problem of the NSS of an UMTS wireless cellular network. 2.1. The assumptions We make the following assumptions concerning the organization of the UMTS network (see Fig. 1): (A1) each node B is connected to exactly one RNC; (A2) each RNC is connected to exactly one MSC; (A3) each RNC is connected to exactly one SGSN; (A4) each MSC is connected to the public switched telephone network (PSTN); (A5) each SGSN is connected to an Internet protocol (IP) based data network; (A6) the number of links connected to a RNC cannot exceed the maximum number of interfaces that can be installed in that RNC; (A7) the sum of the capacities of the node Bs connected to a RNC cannot exceed its switch fabric capacity (in circuit and in bps); (A8) the number of links connected to a MSC cannot exceed the maximum number of interfaces
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that can be installed in that MSC; (A9) the sum of the capacities of the links (in circuit) connected to a MSC cannot exceed its switch fabric capacity (in circuit); (A10) the number of links connected to a SGSN cannot exceed the maximum number of interfaces that can be installed in that SGSN; (A11) the sum of the capacities of the links (in bps) connected to a SGSN cannot exceed its switch fabric capacity (in bps); (A12) at most one RNC can be installed at a RNC site; (A13) at most one MSC can be installed at a MSC site; (A14) at most one SGSN can be installed at a SGSN site. We make the following assumptions concerning the expansion of the network: (A15) each link form a RNC to a MSC (or a SGSN) in the current network may be kept in place or removed from the current network; (A16) each RNC, MSC and SGSN in the current network may not be moved to another site or removed from the network; (A17) new RNCs, MSCs and SGSNs may be installed at new potential sites; (A18) the network topology is preserved in the expanded network. Finally, we suppose that the following information is known: (I1) the location of node Bs in the updated BSS; (I2) the location of facilities forming the current network; (I3) the estimate tra$c (in erlang) for each CoS between the node Bs, between the node Bs, and the PSTN and between the node Bs and the IP data network; (I4) the estimate handover tra$c (in erlang) for each CoS between the node Bs; (I5) the location of new potential RNC sites, MSC sites and SGSN sites; (I6) the di
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(in circuit) of a MSC of type s ∈ S and s the maximum number of interfaces that can be installed in a MSC of type s ∈ S to connect the RNCs); T , the set of SGSN types (where t is the switch fabric capacity (in bps) of a SGSN of type t ∈ T and t the maximum number of interfaces that can be installed in a SGSN of type t ∈ T to connect the RNCs) and C, the set of CoS (where c is the rate (in bps) of a connection of class c ∈ C). 2.2.2. Decision variables Let uij be a 0 –1 variable such that uij = 1 if and only if the node B i ∈ I is connected to a RNC installed at site j ∈ J ; vjk , a 0 –1 variable such that vjk = 1 if and only if a RNC installed at site j ∈ J is connected to a MSC installed at site k ∈ K and vjkm , the number of links of type m ∈ M connecting the RNC installed at site j ∈ J to the MSC installed at site k ∈ K. Let wjl be a 0 –1 variable such that wjl = 1 if and only if a RNC installed at site j ∈ J is connected to a SGSN installed at site l ∈ L and wjln , the number of links of type n ∈ N connecting the RNC installed at site j ∈ J to the SGSN installed at site l ∈ L. Finally, let xjr be a 0 –1 variable such that xjs = 1 if and only if a RNC of type r ∈ R is installed at RNC site j ∈ J ; yks , a 0 –1 variable such that ykt = 1 if and only if a MSC of type s ∈ S is installed at site k ∈ K and zlt , a 0 –1 variable such that zlu = 1 if and only if a SGSN of type t ∈ T is installed at site l ∈ L. 2.2.3. Tra6c variables Let fijcod (fjicod ) be the tra$c (in erlang) of class c ∈ C on the link from node B i ∈ I to site j ∈ J cod cod (fkj ), the (from site j ∈ J to node B i ∈ I ) originating from o ∈ O and destinate to d ∈ D; fjk tra$c (in erlang) of class c ∈ C on the link from site j ∈ J to site k ∈ K (from site k ∈ K to site cod j ∈ J ) originating from o ∈ O and destinate to d ∈ D and fk;cod PSTN (fPSTN; k ), the tra$c (in erlang) of class c ∈ C on the link from site k ∈ K to the PSTN (from the PSTN to site k ∈ K) originating from o ∈ O and destinate to d ∈ D. Let fjlcod (fljcod ) be the tra$c (in erlang) of class c ∈ C on the link from site j ∈ J to site l ∈ L (from site l ∈ L to site j ∈ J ) originating from o ∈ O and destinate to cod d ∈ D and fl;cod IP (fIP; l ), the tra$c (in erlang) of class c ∈ C on the link from site l ∈ L to the IP data network (from site the IP data network to site l ∈ L) originating from o ∈ O and destinate to d ∈ D. Finally, let hcod be the tra$c (in erlang) of class c ∈ C blocked at the network level originating from o ∈ O and destinate to d ∈ D. 2.2.4. Tra6c parameters Let gcod be the tra$c (in erlang) of class c ∈ C originating from o ∈ O and destinate to d ∈ D; cii d G , the handover tra$c (in erlang) from node B i ∈ I to i ∈ I for communications of class c ∈ C destinate to d ∈ D and G coii , the handover tra$c (in erlang) form node B i ∈ I to i ∈ I for communications of class c ∈ C originating from o ∈ O. 2.2.5. Cost parameters Let aqij be the link and interface costs (including the installation cost) for connecting node B i ∈ Iq to a RNC installed at site j ∈ J ; amjk , the link and interface costs (including the installation cost) for connecting a RNC installed at site j ∈ J to a MSC installed at site k ∈ K through a link and interfaces of type m ∈ M and Amjk , the cost of removing the link in the current network connecting a RNC installed at site j ∈ J to a MSC installed at site k ∈ K through a link and interfaces of type
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m ∈ M . Let anjl be the link and interface costs (including the installation cost) for connecting a RNC installed at site j ∈ J to a SGSN installed at site l ∈ L through a link and interfaces of type n ∈ N and Anjl , the cost of removing the link in the current network connecting a RNC installed at site j ∈ J to a MSC installed at site l ∈ L through a link and interfaces of type n ∈ N . Let br be the cost of a RNC of type r ∈ R (including the installation cost); cs the cost of a MSC of type s ∈ S (including the installation cost); dt , the cost of a SGSN of type t ∈ T (including the installation cost) and ec , the cost of loosing a connection of class c ∈ C in the network per erlang during one planning period (de6ned by the network planner). 2.3. The cost function In this subsection, we formulate the cost function representing the total cost of the NSS expansion process, including the cost of new facilities, the cost of installing them and removing links. In order to formulate this function, we need to represent the current network. In this paper, we identify the current network by the decision variables overlined. For instance, if node B i ∈ I is connected to the RNC installed at site j ∈ J in the current network, then uL ij = 1, and if the RNC installed at site j ∈ J is of type s ∈ S, then xLsj = 1. The cost function is composed of three terms: the cost of the links and interfaces, the cost of the nodes (i.e., the RNCs, MSCs and SGSNs) and the cost of the blocked tra$c. The cost of the links and interfaces, noted CL (u; v; w), given by the following equation, includes the cost of the new links and interfaces (including the installation cost) and the cost of removing RNC to MSC links and RNC to SGSN links from the current network. q CL (u; v; w) = aij uij + (amjk (vjkm − vLmjk )+ + Amjk (vLmjk − vjkm )+ ) q ∈ Q i ∈ Iq j ∈ J
+
j ∈ J l∈ L n ∈ N
j ∈ J k ∈ K m∈ M
(anjl (wjln − wL njl )+ + Anjl (wL njl − wjln )+ );
(1)
where (x)+ = max{0; x}. The cost of the nodes, noted CN (x; y; z), given by the following equation, includes the cost of the new RNCs, MSCs and SGSNs (including the installation cost): + s s + t t + CN (x; y; z) = xjr − xLrj + yk − yL tk + zl − zLtl : br c d (2) r ∈R
j ∈J
s ∈S
k ∈K
The cost of the blocked tra$c, noted CT (h), is CT (h) = ec hcod :
t ∈T
l∈ L
(3)
c ∈ C o∈ O d∈ D
2.4. The model The model for the expansion problem of the NSS of an UMTS wireless cellular network, noted EPN, can now be given.
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EPN: min
f;h;u;v;w;x;y;z
CL (u; v; x) + CN (x; y; z) + CT (h)
s:t:
(4)
Node B assignment constraints: uij = 1 (i ∈ I ):
(5)
j ∈J
RNC assignment constraints: vjk = xjr (j ∈ J );
(6)
r ∈R
k ∈K
wjl =
r ∈R
l∈ L
xjr
(j ∈ J ):
(7)
RNC-type uniqueness constraints: xjr 6 1 (j ∈ J ):
(8)
MSC-type uniqueness constraints: yks 6 1 (k ∈ K):
(9)
SGSN-type uniqueness constraints: zlt 6 1 (l ∈ L)
(10)
r ∈R
s ∈S
t ∈T
Node B capacity constraints: coi cid fji + fij 6 q uij c ∈C
o∈ O
c
c ∈C
o∈ O
d∈ D
fjicoi +
(i ∈ Iq ; j ∈ J; q ∈ Q);
fijcid
6 q uij
(i ∈ Iq ; j ∈ J; q ∈ Q):
i ∈ Iq
m∈ M k ∈ K
vjkm +
(12)
d∈ D
RNC capacity constraints (at the interface level): q uij 6 r1 xjr (j ∈ J ); q ∈Q
(11)
r ∈R
n ∈ N l∈ L
wjln 6
r ∈R
r2 xjr
(j ∈ J ):
(13) (14)
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RNC capacity constraints (at the switch fabric level): q uij 6 r xjr (j ∈ J ); i ∈ Iq
q ∈Q
q
uij 6
i ∈ Iq
q ∈Q
(15)
r ∈R
r ∈R
r xjr
(j ∈ J ):
(16)
MSC capacity constraints (at the interface level): vjkm 6 s yks (k ∈ K):
(17)
MSC capacity constraints (at the switch fabric level): m vjkm 6 s yks (k ∈ K):
(18)
SGSN capacity constraints (at the interface level): wjln 6 t zlt (l ∈ L):
(19)
SGSN capacity constraints (at the switch fabric level): n wjln 6 t zlt (l ∈ L):
(20)
RNC-MSC link capacity constraints: cod fjk 6 m vjkm (j ∈ J; k ∈ K);
(21)
m∈ M j ∈ J
m∈ M
s ∈S
j ∈J
s ∈S
n∈ N j ∈ J
n∈ N
t ∈T
j ∈J
t ∈T
m∈ M
c ∈ C o∈ O d∈ D
c ∈ C o∈ O d∈ D
cod fkj 6
m∈ M
m vjkm
(j ∈ J; k ∈ K);
RNC-SGSN link capacity constraints: c fjlcod 6 n wjln (j ∈ J; l ∈ L); c ∈C
c ∈C
o∈ O d ∈ D
(23)
n∈ N
o∈ O d ∈ D
c
(22)
fljcod
6
n∈ N
n wjln
(j ∈ J; l ∈ L):
(24)
CAC and CoS priority constraints: CAC and CoS priority constraints :
(25)
Tra6c =ow conservation constraints: Tra$c Jow conservation constraints :
(26)
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Additional constraints: vjk max{r2 } ¿ vjkm r ∈R
vjk 6
m∈ M
(j ∈ J; k ∈ K);
r ∈R
n∈ N
(27)
m∈ M
vjkm
wjl max{r2 } ¿ wjl 6
(j ∈ J; k ∈ K);
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n∈ N
wjln
wjln
(28)
(j ∈ J; l ∈ L);
(29)
(j ∈ J; l ∈ L):
(30)
Expansion constraints: uij ¿ uL ij xjr ¿ xLrj yks ¿ yL sk zlt ¿ zLtl
(i ∈ I; j ∈ J );
(31)
(j ∈ J; r ∈ R);
(32)
(k ∈ K; s ∈ S);
(33)
(l ∈ L; t ∈ T ):
(34)
Nonegativity constraints: 2|C OD|(|J |(|I |+|K |+|L|)+|K |+|L|)
f ∈ R+
:
(35)
Integrality constraints: vjk ∈ B; vjkm ∈ N
(j ∈ J; k ∈ K; m ∈ M );
wjl ∈ B; wjln ∈ N
(j ∈ J; l ∈ L; n ∈ N );
u ∈ B|I J | ;
x ∈ B | J R| ;
y ∈ B |K S | ;
z ∈ B | L T | :
(36)
The objective function (4) of EPN, as mentioned before, is composed of three terms, representing the expansion cost of the NSS. Constraints (5) are node B assignment constraints that require each node B to be connected to exactly one RNC and constraints (6) and (7) are RNC assignment constraints that require each RNC to be connected to exactly one MSC and exactly one SGSN. The RNC-type uniqueness constraints (8) impose that at most one RNC type be installed at site j ∈ J , MSC-type uniqueness constraints (9) impose that at most one MSC type be installed at site k ∈ K and SGSN-type uniqueness constraints (10) impose that at most one SGSN type be installed at site l ∈ L. Constraints (11) necessitate the total tra$c (in erlang) from node B i ∈ I be less or equal to its capacity (in circuit) and constraints (12) impose the total tra$c (in bps) from node B i ∈ I be less or equal to its capacity (in bps). Constraints (13) necessitate the total number of links from the node Bs to the RNC type installed at site j ∈ J be less than or equal to the maximum number of interfaces that can be installed in that RNC for the node Bs and constraints (14) impose the total number of links from the RNC type installed at site j ∈ J to the MSCs and SGSNs be less than or equal to the maximum number of interfaces that can be installed in that RNC for the MSCs and SGSNs. Constraints (15) dictate the sum of the capacities (in circuit) of the node Bs connected to the RNC type installed at site j ∈ J be less than or equal to its switch fabric capacity (in circuit)
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and constraints (16) impose the sum of the rates (in bps) of the node Bs connected to the RNC type installed at site j ∈ J be less than or equal to its switch fabric capacity (in bps). Constraints (17) necessitate the total number of links from the RNCs to the MSC type installed at site k ∈ K be less than or equal to the maximum number of interfaces that can be installed in that MSC and constraints (18) dictate the total capacity of the links (in circuit) from the RNCs to the MSC type installed at site k ∈ K be less than or equal to the switch fabric capacity (in circuit) of that MSC. Constraints (19) necessitate the total number of links from the RNCs to the SGSN type installed at site l ∈ L be less than or equal to the maximum number of interfaces that can be installed in that SGSN and constraints (20) dictate the total capacity of the links (in circuit) from the RNCs to the SGSN type installed at site l ∈ L be less than or equal to the switch fabric capacity (in circuit) of that SGSN. Constraints (21) and (22) are the RNC to MSC link capacity constraints (in circuit) and constraints (23) and (24) are the RNC to SGSN link capacity constraints (in bps). Constraints (25) detail how the connection admission is performed in the NSS and the priorities are applied between the class of services. In order to have a model that can be adapted to any standard and proprietary CAC algorithms and CoS prioritization schemes, these constraints are not speci6ed directly in the model. For instance, let C = {c0 ; : : : ; c|C |−1 } such that c0 is the class for the voice service and c1 to c|C |−1 the classes for the data services such that c1 is the 6rst priority class, c2 the second priority class, etc. As a result, ec0 ¿ · · · ¿ ec|C |−1 and constraints (25) can be formulated as follows. For the node B to RNC links, (37) fijc0 od + fjic0 od 6 q ; o∈ O d ∈ D
o∈ O d ∈ D
fijc0 od + fjic0 od ; fijc1 od + fjic1 od 6 q −
(38)
o∈ O d∈ D
··· o∈ O d ∈ D
c
(fij|C |−1
od
c
od
+ fji|C |−1 ) 6 q −
|C |−2
l=0 o∈O d∈D
(fijcl od + fjicl od );
(39)
for all i ∈ Iq , q ∈ Q and j ∈ J . Akin constraints can be written down for the other links (i.e., the RNC to MSC links and the RNC to SGSN links). Constraints (26), detailed below, are tra$c Jow conservation constraints. cod g + G ci od − hcod if o = i cod fij = (i ∈ I; c ∈ C; o ∈ O; d ∈ D); (40) i ∈I \{o} 0 otherwise j ∈J cod g + G ci od − hcod if d = i cod fji = (i ∈ I; c ∈ C; o ∈ O; d ∈ D); (41) i ∈I \{o} 0 otherwise j ∈J cod fijcod − fjk = 0 (j ∈ J; c = c0 ; o ∈ O; d ∈ D); (42) i ∈I
k ∈K
S. Chamberland / Computers & Operations Research 31 (2004) 1769 – 1791
cod fkj −
k ∈K
i ∈I
fijcod −
j ∈J
j ∈J
i ∈I
fjicod = 0
(j ∈ J; c = c0 ; o ∈ O);
(43)
fjlcod = 0
(j ∈ J; c ∈ C \ {c0 }; o ∈ O; d ∈ D);
(44)
fjicod = 0
(j ∈ J; c ∈ C \ {c0 }; o ∈ O; d ∈ D);
(45)
l∈ L
fljcod −
l∈ L
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cod fjk −
fjlcod −
i ∈I
j ∈J
j ∈J
fk;cod PSTN −
k ∈K
cod cod cod fkj + fPSTN; k − fk; PSTN = 0
cod cod fljcod + fIP; l − fl; IP = 0
k ∈K
cod fPSTN; k
cod g = −gcod 0
cod g cod cod fl; IP − fIP; l = −gcod l∈ L l∈ L 0
(k ∈ K; c = c0 ; o ∈ O; d ∈ D);
(46)
(l ∈ L; c ∈ C \ {c0 }; o ∈ O; d ∈ D):
(47)
if d = PSTN if o = PSTN
(c ∈ C; o ∈ O; d ∈ D);
(48)
otherwise
if d = IP if o = IP
(c ∈ C; o ∈ O; d ∈ D):
(49)
otherwise
Constraints (27) and (28) impose for all j ∈ J and k ∈ K, vjk = 1 if and only if m∈M vjkm ¿ 1 and constraints (29) and (30) dictate for all j ∈ J and l ∈ L, wjl = 1 if and only if n∈N wjln ¿ 1. Constraints (31) dictate the node B to RNC links installed in the current network to stay in place in the expanded network, constraints (32)–(34) impose each RNC, MSC and SGSN installed in the current network to stay in place in the expanded network and, 6nally, constraints (35) are nonnegativity constraints and constraints (36) the integrality constraints. We remind to the reader that the objective function of EPN was formulated to reach a compromise between the cost of the NSS and the cost of the blocked tra$c. However, if the network planner wants to consider a blocking probability limit at the NSS level, EPN can be easily adapted by including a new set of constraints. For instance, if a blocking probability limit of 0.1% is wanted per origin–destination, the following set of constraints can be added in the EPN formulation: c ∈C
gcod +
hcod 6 0:1 ciod + G coid ) i∈I (G
(o ∈ O; d ∈ D):
Since EPN is NP-hard (transformation from the uncapacity facility location problem), in the rest of this paper, we concentrate our e
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3. The tabu search algorithm In this section, we propose a tabu search (TS) algorithm for EPN, called TS-EPN. An introduction to tabu search can be found in [15]. The following notation is used in the presentation of TS-EPN. Let 4j be the state of the site j ∈ J such that 4j = 0 if there is no RNC installed at site j and 4j = r (for r ∈ R) if a RNC of type r is installed at site j. Similarly, let 4k be the state of the site k ∈ K such that 4k = 0 if there is no MSC installed at site k and 4k = s (for s ∈ S) if a MSC of type s is installed at site k and 6nally, let 4l be the state of the site l ∈ L such that 4l = 0 if there is no SGSN installed at site l and 4l = t (for t ∈ T ) if a SGSN of type t is installed at site l. Let y(), y() and z() be the vectors x, y and z when the state vector of all sites = {4j }j∈J {4k }k ∈K {4l }l∈L is 6xed. In the next subsection, we propose a decomposition approach to solve EPN(), i.e., the model EPN when the state vector is 6xed. 3.1. Solving EPN() When vectors x, y and z are 6xed, EPN can be decomposed into two subproblems. The 6rst subproblem, noted EPN(x; y; z), is given below. EPN(x; y; z): q aij uij min u q ∈ Q i ∈ Iq j ∈ J (50) s:t:
(5); (13); (16); (31) and u ∈ B|I J | :
(51)
The purpose of this subproblem is to connect the node Bs to the RNCs while respecting assignment, RNC degree and capacity constraints. Since this subproblem is NP-hard (transformation from the knapsack problem) and we may have to solve it several thousands instances of it during the tabu search procedure, we propose a heuristic to 6nd solutions quickly. This heuristic, called heuristic for the 6rst subproblem (HFS), is presented below. Heuristic HFS. Step 1: Order the elements in Q such that q0 ¿ q1 ¿ · · · ¿ q|Q|−1 and q0 ≺ q1 ≺ : : : ≺ q|Q|−1 . Step 2: For q := q0 to q|Q|−1 do Solve the following problem, called EPN q (u; x; y; z), that consists to connect the node Bs in Iq to the RNCs considering that the node Bs in q ∈Q:q ≺q Iq are already connected to RNCs. EPNq (u; x; y; z): q min (52) aij uij {uij :i∈Iq ; j∈J }
i ∈ Iq j ∈ J
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s:t:
uij = 1
(i ∈ Iq );
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(53)
j ∈J
i ∈ Iq
i ∈ Iq
i ∈ Iq
1 r r q uij 6 q 1 x j − uij r ∈R q ∈Q:q ≺q i∈I
(j ∈ J );
(54)
1 r r q uij 6 q xj − uij r ∈R q ∈Q:q ≺q i∈I
(j ∈ J );
(55)
1 r r q uij 6 x − u ij j q r ∈R q ∈Q:q ≺q i∈I
(j ∈ J );
(56)
q
q
q
uij ∈ R+
(i ∈ Iq ; j ∈ N ):
(57)
This subproblem is an instance of the linear assignment problem; to solve it, we use the shortest augmenting path algorithm LAPJV of Jonker and Volgenant [16]. Step 3: Return the vector u and the cost of the subproblem using (50). The second subproblem, noted EPN(u; x; y; z), is given below. EPN(u; x; y; z): min (amjk (vjkm − vLmjk )+ + Amjk (vLmjk − vjkm )+ ) f;h;v;w
j ∈ J k ∈ K m∈ M
+
j ∈ J l∈ L n ∈ N
s:t:
(anjl (wjln − wL njl )+ + Anjl (wL njl − wjln )+ ) + CT (h)
(58)
(6); (7); (11); (12); (14); (17)–(30) and 2|C OD|(|J |(|I |+|K |+|L|)+|K |+|L|)
f ∈ R+
vjk ∈ B; vjkm ∈ N
(j ∈ J; k ∈ K; m ∈ M );
wjl ∈ B; wjln ∈ N
(j ∈ J; l ∈ L; n ∈ N ):
(59)
(60)
The purpose of this subproblem is to connect the RNCs to the MSCs and SGSNs while respecting assignment constraints, RNC degree constraints, MSC degree and capacity constraints, SGSN degree and capacity constraints, link capacity constraints and tra$c Jow conservation constraints. Since this subproblem is NP-hard (transformation from the knapsack problem), we propose a heuristic to 6nd solutions quickly. This heuristic, called Heuristic for the second subproblem (HSS), is presented below.
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Heuristic HSS. Step 1: For all j ∈ J do 1.1. Calculate Tra$c (i; j; c) de6ned as the tra$c (in erlang) of class c ∈ C from node B i ∈ I to RNC j that would be switched by a MSC or a SGSN. Therefore, for all i ∈ I and c ∈ C, if uij = 1 set gcid + gcoi + G ci id Tra$c(i; j; c) := d∈ D
i ∈ I d∈ D
o∈ O
otherwise set Tra$c(i; j; c) := 0. 1.2. Calculate BlockedTra$c(i; j; c) de6ned as the tra$c (in erlang) of class c ∈ C blocked from node B i ∈ I to RNC j using constraints (25). 1.3. Calculate Tra$cMSC (j) de6ned as the tra$c (in erlang) passing through the RNC j that should be switched by a MSC and Tra$cSGSN (j), the tra$c (in bps) passing through the RNC j that should be switched by a SGSN. Therefore, (Tra$c(i; j; c0 ) − BlockedTra$c(i; j; c0 )); Tra$cMSC (j) := i∈I :uij =1
Tra$cSGSN (j) :=
c (Tra$c(i; j; c) − BlockedTra$c(i; j; c)):
i∈I :uij =1 c∈C \{c0 }
1.4. Calculate MinLinksMSC (j) de6ned as the minimum number of links needed from the RNC j to a MSC and MinLinksMSC (j), the minimum number of links needed from the RNC j to a SGSN. Therefore, Tra$cMSC (j) ; MinLinksMSC (j) := maxm∈M { m } Tra$cSGSN (j) MinLinksSGSN (j) := maxn∈N { n } and set MinLinks(j) := MinLinksMSC (j) + MinLinksSGSN (j). 1.5. If MinLinks(j) 6 r ∈R r2 xjr go to Step 1.6. Otherwise, stop, the subproblem is not feasible. 1.6. Determine by enumeration the number of links of each type NumLinksMSC (j; m) for all m ∈ M , such that the total number of links is MinLinksMSC (j) and the total capacity of these links (in erlang), noted TotalLinkCapacityMSC (j), is minimum but greater or equal to Tra$cMSC (j). 1.7. Calculate NumLinksInstallMSC (j; k; m) de6ned as the number of links of type m ∈ M to install if RNC j is connected to MSC k ∈ K and NumLinksRemoveMSC (j; k; m), the number of links of type m ∈ M to remove if RNC j is connected to MSC k ∈ K. Therefore, for all k ∈ K and m ∈ M , set NumLinksInstallMSC (j; k; m) := (NumLinksMSC (j; m) − vLmjk )+ NumLinksRemoveMSC (j; k; m) :=
k ∈K \{k }
vLmjk :
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1.8. Determine by enumeration the number of links of each type NumLinksSGSN (j; n) for all n ∈ N , such that the total number of links is MinLinksSGSN (j) and the total capacity of these links (in bps), noted TotalLinkCapacitySGSN (j), is minimum but greater or equal to Tra$cSGSN (j). 1.9. Calculate NumLinksInstallSGSN (j; l; n) de6ned as the number of links of type n ∈ N to install if RNC j is connected to SGSN l ∈ L and NumLinksRemoveSGSN (j; l; n), the number of links of type n ∈ N to remove if RNC j is connected to SGSN l ∈ L. Therefore, for all l ∈ L and n ∈ N , set NumLinksInstallSGSN (j; l; n) := (NumLinksSGSN (j; n) − wL njl )+
NumLinksRemoveSGSN (j; l; n) :=
wL njl :
l ∈L\{l}
Step 2: Let 51 ; 52 ; : : : ; 5|6| be the di
s:t:
j ∈JMSC (p) k ∈K
vjk = 1
m∈ M
(j ∈ JMSC (p));
k ∈K
1 s s vjk 6 yj − ’ p j ∈J s ∈S MSC(p)
j ∈JMSC(p)
1 vjk 6 s yjs − 5p s ∈ S
p −1 j ∈∪p =1 JMSC(p )
(62) MinLinksMSC (j)vjk
(k ∈ K);
TotalLinkCapacityMSC (j)vjk
(63)
(k ∈ K);
(64)
p −1 j ∈∪p =1 JMSC(p )
vjk ∈ R+ (j ∈ JMSC (p); k ∈ K);
(65)
where aLmjk := amjk NumLinksInstallMSC (j; k; m); AL mjk := Amjk NumLinksRemoveMSC (j; k; m): This subproblem is an instance of the linear assignment problem; to solve it, we use the shortest augmenting path algorithm LAPJV of Jonker and Volgenant [16].
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Step 4: Let 91 ; 92 ; : : : ; 9|:| be the di
s:t:
j ∈JSGSN (p) l∈L
wjl = 1
n∈ N
(j ∈ JSGSN (p));
l∈ L
j ∈JSGSN (p)
1 t t wjl 6 z − $p t ∈ T l
j ∈JSGSN (p)
1 wjl 6 t zlt − 9p t ∈ T
wjl ∈ R+
(j ∈ JSGSN (p); l ∈ L);
p −1 j ∈∪p =1 JSGSN(p )
(67) MinLinksSGSN (j)wjl (l ∈ L);
(68)
TotalLinkCapacitySGSN (j)wjl (l ∈ L);
(69)
p −1 j ∈∪p =1 JSGSN(p )
(70)
where aLnjl := anjl NumLinksInstallSGSN (j; l; n); AL njl := Anjl NumLinksRemoveSGSN (j; l; n): This subproblem is an instance of the linear assignment problem; to solve it, we use the shortest augmenting path algorithm LAPJV of Jonker and Volgenant [16]. Step 6: For all j ∈ J , k ∈ K and m ∈ M set vjkm := NumLinksMSC (j; m)vjk and for all j ∈ J , l ∈ L and n ∈ N set wjln := NumLinksSGSN (j; n)wjl . Step 7: Return the vectors v and w and the cost of the subproblem using (58). 3.2. The tabu search algorithm Each move of tabu search consists of modifying the state of a given site in the current solution (respecting constraints (32)–(34)). At each iteration of the search, we determine the best move (among the |J R| + |K S| + |L T | possible moves) while taking into account the tabus as well as the aspiration criterion. The chosen site is declared tabu for a number of iterations that is randomly determined according to a uniform discrete distribution on a speci6ed interval.
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The aspiration criterion states that if the use of tabu site allows us to discover a solution better than any other found so far, we may remove the tabu from this site. We now proceed to a detailed description of TS-EPN. Algorithm TS-EPN. Step 1: (Initial solution) Find an initial solution by solving EPN() where vector is determined as follows. • for all j ∈ J , if for a value of r ∈ R, xLrj = 1 set 4j := r, otherwise set 4j := arg maxr ∈R {r }, • for all k ∈ K, if for a value of s ∈ S, yL sk = 1 set 4k := s, otherwise set 4k := arg maxs∈S {s }, • for all l ∈ L, if for a value of t ∈ T , zLtl = 1 set 4l := t, otherwise set 4l := arg maxt ∈T {t }. Repeat Steps 2–3 for Max Iter iterations. Step 2 (Exploring the neighborhood): If the iteration count is a multiple of 10, apply the perturbation method (presented in the next subsection) to the current solution. Otherwise, do Steps 2.1 and 2.2. 2.1. Determine the best move while taking into account the tabus, the aspiration criterion and constraints (32)–(34). For each move → , which modi6es the state of a given site in the current solution, we solve EPN( ) as described in Section 3.1. The cost of a solution is given by the objective function (4). 2.2. Determine the number of iterations according to a uniform distribution on a speci6ed interval for which the chosen site is tabu. Step 3 (Best solution update): If the cost of the current solution is less than the cost of the best solution found so far, update this best solution. In the algorithm TS-EPN, the neighborhood can be fully explored or randomly explored. In the latter case, the percentage of the neighborhood explored should be speci6ed in the algorithm. Other parameters should be selected, i.e., the number of iterations (i.e., the Max Iter parameter) and the interval of the number of iteration that a site can be tabu. 3.3. The perturbation method We propose a perturbation method for visiting several local minima in few iterations of the search process. The proposed perturbation method, performed once every 10 iterations, consists of choosing the minimum cost solution obtained from the current solution by 1. the elimination of a new node (i.e., a new RNC, MSC or SGSN); 2. the installation of a new node at an empty site (i.e., a new candidate site such that its state is zero); 3. the transfer of a new node installed from a site to an empty site. Each site so that its state is changed by the perturbation method is declared tabu for a number of iterations determined in the same way as it is done in Step 2.2 of TS-EPN.
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Table 1 Features of the node B types
Capacity (circuits) Capacity (Mbps) Number of ATM interfaces
Type A
Type B
Type C
100 120 1 × OC-3
200 240 2 × OC-3
400 480 1 × OC-12
Table 2 Costs of the RNC types (including the installation costs)
Switch fabric capacity (circuits) Switch fabric capacity (Gbps) Maximum number of node B interfaces Maximum number of MSC/SGSN interfaces Cost ($)
Type A
Type B
Type C
1000 2 10 15 50,000
2500 5 20 30 90,000
5000 10 40 60 120,000
Table 3 Costs of the MSC types (including the installation costs)
Switch fabric capacity (circuits) Maximum number of interfaces Cost ($)
Type A
Type B
Type C
100,000 50 200,000
200,000 100 350,000
300,000 150 500,000
Type A
Type B
Type C
20 16 40,000
40 32 60,000
80 64 80,000
Table 4 Costs of the SGSN types (including the installation costs)
Switch fabric capacity (Gbps) Maximum number of interfaces Cost($)
4. Numerical results In this section, we present a study of the performance of the proposed heuristics. The algorithm TS-EPN was implemented in the C language on a SunFire 4800 workstation (4 CPU, 900 MHz and 4 GB of RAM). For the tests, three node B types, three RNC types, three MSC types and three SGSN types are used. Their features are presented respectively in Tables 1–4. Moreover, asynchronous transfer mode (ATM) OC-3 and OC-12 links are used to connect the node Bs to the RNCs, DS-3 links are used to connect the RNCs to the MSCs and gigabit Ethernet (GE) links to
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Table 5 Costs of the links (including the installation costs) Link type
Capacity
Cost ($/km)
DS-3 OC-3 OC-12 GE
2688 circuits 149:760 Mbps 594:432 Mbps 1 Gbps
2500 1500 4000 4000
Table 6 Costs of the interface types (including the installation costs) Interface type
Cost ($)
DS-3 OC-3 OC-12 GE
1500 2000 4500 2000
connect the RNCs to the SGSNs. The link costs are presented in Table 5 and the interface costs in Table 6. Finally, the cost of removing a link is 100$=km. Three classes of service are considered for the tests. The 6rst class (class 1) is used for voice service (using a 16 kbps compression), the class 2 is used for the 144 kbps data services and, the class 3 for the 384 kbps data service. The cost of loosing a connection of class 1, 2 and 3 during per erlang per planning period is 180$, 200$ and 220$ respectively. For the tests, the Max Iter parameter of the TS-EPN heuristic is set to 100 and the interval of the number of iteration that a site can be tabu to [5,10] because the best solutions was found within 100 iterations and with this interval. Moreover, at each iteration of TS-EPN, the neighborhood is fully explored. TS-EPN results were compared to a lower bound (denoted LB) obtained as follows. The bound LB was obtained by using the CPLEX Mixed Integer Solver [17] to solve EPN without constraints (11), (12), (21)–(26) and integrality constraints on v and w variables. The default settings of CPLEX are used and the branch-and-bound node limit was set to 100,000 and the upper limit on the size of the tree to 100 MB. For the evaluation of the heuristic performance for network design, 40 design problems are generated randomly as follows. |I | points corresponding to node Bs’ locations, |J | points corresponding to the potential RNC sites, |K| points corresponding to the potential MSC sites and |L| points corresponding to the potential SGSN sites were generated in a square region of length of 100 km following a uniform distribution. The type of each node B is selected randomly among the three node B types considered (see Table 1). The call tra$c between each pair of node Bs and between the node Bs and the PSTN and IP data network is generated randomly in the interval [0; 5] erlang,
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following a uniform distribution, for each class of service. Finally, the hanover tra$c between each pair of node Bs is generated randomly in the interval [0; 1] erlang, following a uniform distribution, for each destination and class of service. The results for the design problems are presented in Table 7 . In this table, columns 1– 4 present respectively the number of node Bs, the number of potential RNC sites, the number of potential MSC sites and the number of potential SGSN sites. Columns 5 –7 present the solution found by algorithm TS-EPN, the CPU execution time to 6nd it and the GAP that indicates the percentage gap between the heuristic solution and the lower bound LB (with respect to the value of the lower bound). The following observations can be made from Table 7. First, the heuristic TS-EPN 6nds solutions close to optimality for the design problems considered (within 2.65%, on average, from the lower bound) and the mean CPU time is 758 s. Moreover, the reader should point out that the heuristic 6nds the optimal solution four times (i.e., on average, TS-EPN 6nd the optimal solution 10% of the time for the design problems considered). For the evaluation of the heuristic performance for network expansion, 40 expansion problems are generated randomly. The results are presented in Table 8. In this table, columns 1 to 4 present respectively the number of node Bs, the number RNCs, the number MSCs and the number SGSNs in the initial network. Columns 5 –8 present respectively the number of new node Bs, the number of new potential RNC sites, the number of new potential MSC sites and the number of new potential SGSN sites. Columns 9 –11 present the solution found by algorithm TS-EPN, the CPU execution time to 6nd it and the GAP that indicates the percentage gap between the heuristic solution and the lower bound LB (with respect to the value of the lower bound). The following observations can be made from Table 8. First, the heuristic TS-EPN 6nds solutions close to optimality for the expansion problems considered (within 8.07%, on average, from the lower bound) and the mean CPU time is 2818 s. Moreover, the heuristic 6nds the optimal solution three times. For all solutions found having a gap greater than 2%, more than one link is installed between at least one RNC to a MSC or a SGSN. However, since the Jow variables and constraints are not considered to 6nd the value of LB, a unique link is used from each RNC to the MSCs and a unique link is used from each RNC to the SGSNs. We have tried to improve the value of LB by considering the Jow variables and constraints in the LB formulation but we did not succeed to solve it using CPLEX. 5. Conclusions In this paper we have studied the problem of expanding UMTS cellular wireless networks and proposed an optimization model for the location of new RNCs, MSCs and SGSNs, selecting their types, updating the network connectivity, while considering the call and handover blocking. After de6ning network elements of such networks and making some realistic assumptions about their organization, we have presented a mathematical formulation of this planning problem. A tabu search algorithm was proposed to 6nd good solutions. All the results were compared to a proposed lower bound obtained by relaxing a subset of the original constraints in the model. These results have shown that the tabu based heuristic produces
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Table 7 Computational results for the design problems Number of node Bs
Number of potential RNC sites
Number of potential MSC sites
Number of potential SGSN sites
TS-EPN OBJ
CPU
GAP
20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100
5 5 5 5 5 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20 5 5 5 5 5 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
2050.3 4134.3 6053.0 9063.4 10026.4 2257.5 3863.9 5663.1 6879.1 7290.8 2295.7 3971.3 5651.3 6064.7 6669.7 2180.1 3170.4 4532.2 5594.8 7246.4 2134.5 3985.8 7014.2 8082.3 8438.6 2180.0 3862.5 4454.5 6318.6 7790.4 2257.2 3536.9 4792.3 5761.2 7169.6 1788.9 3400.1 4721.2 6111.7 6977.6
10 40 156 498 1006 18 60 225 756 1465 35 88 302 980 1907 69 136 416 1258 2428 41 88 281 933 1814 70 130 382 1192 2257 120 197 516 1450 2755 199 288 677 1790 3301
0.00 2.22 2.17 7.51 3.77 2.26 0.88 1.56 2.94 5.45 0.00 2.06 2.63 3.77 5.03 2.78 0.65 1.65 2.36 5.07 0.19 0.98 3.64 5.95 7.16 0.00 0.54 1.68 5.96 4.76 0.83 0.00 0.59 3.63 3.50 0.97 0.91 3.61 2.96 3.50
5135.9
758
2.65
Mean
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Table 8 Computational results for the expansion problems Initial network
New node Bs and sites
TS-EPN
Number of node Bs
Number of RNCs
Number of MSCs
Number of SGSNs
Number of new node Bs
Number of new potential RNC sites
Number of new potential MSC sites
Number of new potential SGSN sites
OBJ (k$)
CPU (s)
GAP (%)
20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 80 80 80 80 80 80 80 80 100 100 100 100 100 100 100 100
2 4 4 4 3 4 3 3 5 5 6 5 3 5 6 6 4 5 7 8 3 5 6 8 4 8 8 8 4 6 8 10 5 8 10 10 5 8 9 10
1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 1 1 1 2 2 2 1 3 2 1 2 2 1 3 2 2 3 2 3 3 2 2
1 2 2 2 2 3 2 2 3 2 2 3 2 3 3 3 1 2 2 2 2 4 4 4 2 4 4 3 2 3 5 4 2 2 3 3 3 4 4 5
20 20 40 40 60 60 80 80 20 20 40 40 60 60 80 80 20 20 40 40 60 60 80 80 20 20 40 40 60 60 80 80 20 20 40 40 60 60 80 80
5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10
5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10
5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10
1905.1 1923.7 4150.1 2855.4 7055.1 4308.2 6943.2 5518.8 1830.7 1997.8 3744.9 2815.4 7319.8 4306.0 6670.7 5383.6 1432.1 1729.7 3311.7 2908.7 7686.0 4450.8 6753.2 4714.3 1810.8 1498.3 3587.2 2777.8 6056.0 3627.8 5121.3 4513.0 1328.3 1458.6 3096.8 2585.8 7007.3 4174.0 5637.7 4553.9
41 151 154 402 525 1194 1030 2252 155 397 539 1179 1041 2244 1764 3769 539 1182 1060 2266 1778 3803 2808 5944 1066 2233 1825 3809 2837 5940 4245 8744 1806 3693 2850 5921 4181 8791 5984 12563
0.00 0.00 2.55 0.73 5.22 4.80 10.20 3.50 12.30 7.97 12.63 7.44 6.99 6.32 7.90 2.47 6.87 16.93 16.61 9.11 5.61 4.65 13.41 8.57 0.00 8.20 11.38 10.52 3.64 9.93 12.16 5.65 6.43 23.67 7.26 24.70 1.83 3.94 11.00 9.71
4013.7
2818
8.07
Mean
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solutions close to the lower bound for test networks. For the design test problems, the heuristic has found solutions within, on average, 2.65% from the lower bound and for the expansion problem, the heuristic has found solutions within, on average, 8.07% from the lower bound. As a result, the heuristics proposed are recommended for large-size cellular networks with an important number of cells. It would be interesting to improve the lower bound formulation in order to evaluate more e
Computers & Operations Research 31 (2004) 1769 – 1791
www.elsevier.com/locate/dsw
An e$cient heuristic for the expansion problem of cellular wireless networks Steven Chamberland∗ CRT and Computer Engineering Department, Ecole Polytechnique de Montreal, C.P. 6079, Succ. Centre-Ville, Montreal (Quebec), Canada H3C 3A7
Abstract In this paper we propose a model for the expansion problem of the network subsystem (NSS) of a universal mobile telecommunication system (UMTS) wireless cellular network considering an update in the base station subsystem (BSS). The objective is to minimize the expansion cost of the network subsystem while considering network performance (e.g., call and handover blocking). Since the network expansion problem is a generalization of the design problem, the proposed model can also be used for designing networks. In order to 6nd good solutions, we propose a heuristic based on the tabu search principle. Finally, we present a performance analysis of the proposed heuristic. The analysis shows that quasi-optimal solutions are found with the proposed heuristic. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Cellular networks; Topological expansion; Facility location and type selection; Tabu search
1. Introduction Cellular wireless network operators dedicate an important proportion of their budget to acquire, install and maintain the facilities that carry tra$c from cell sites to switches and gateways. These facilities are often leased from local exchange carriers. The pressure to reduce costs adds new urgency to the search for optimized networks, which can minimize the cost of required facilities while satisfying a set of predetermine constraints. Considering the potential augmentation of the number of customers for the services o
Corresponding author. Tel.: +1-514-340-4711 x 2996; fax: +1-514-340-3240. E-mail address: [email protected] (S. Chamberland).
0305-0548/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0305-0548(03)00119-9
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In a typical UMTS network, the area of coverage is geographically divided into cells and the network topology is hierarchically organized in order to reduce costs. Each cell is implemented with a base transceiver station (node B) that contains the radio transceivers providing the radio interface with mobile stations. One or more node Bs are connected to a radio network controller (RNC) that provides a number of functions related to resource and mobility management as well as operation and maintenance for the overall radio network. One or more RNCs are connected to a mobile switching center (MSC) or switch that control call setup, call routing, while performing many other functions provided by a conventional communications switch. Each MSC provides connectivity to the public switched telephone network (PSTN) for the voice services. Moreover, one or more RNCs are connected to a SGSN (serving GPRS (general packet radio service) support node) or gateway. Each SGSN provides connectivity to an Internet protocol (IP) data network. Typically, the expansion of a cellular wireless networks requires: 1. the analysis of the network performance based on, among others, call and handover blocking; 2. the analysis of radio-wave propagation and/or the 6eld topology to identify a set of new cells to improve the network performance; 3. the expansion of the node Bs to RNCs network while taking into account a certain number of constraints including capacity constraints; 4. the location of new RNCs, MSCs and SGSNs and the selection of their types; 5. the update of the 6xed network connectivity considering topological constraints and the network performance. In this paper, we are interested in the expansion problem of the network subsystem (NSS) of an UMTS wireless cellular network considering an update in the base station subsystem (BSS). The proposed model deals with • locating new RNCs, MSCs and SGSNs; • selecting their types; • updating the network connectivity; while considering the call and handover blocking. Furthermore, we suppose that several classes of service (CoS) are available in the network (e.g., one class for voice services and others for data services). The objective is to minimize the cost of the expansion process, including the cost of the equipments and the installation costs. The literature of Engineering and Operations Research contains many articles relating to NSS planning problems. Some NSS planning studies focus only on a portion of the overall problem, such as, the location problem of controllers and switches [2–4], the assignment problem of cell to switches [5–9] and the topological design and link dimensioning problem [10–12]. Other studies concern these problems with additional features such as diversity or survivability constraints [10,13] and multi-period demand [14]. However, the overall NSS planning problem tackle in this paper was not considered so far. The rest of this paper is organized as follows. In Section 2 we present a model for the expansion problem of the NSS of an UMTS wireless cellular network. In Section 3, we present a heuristic based on the tabu search principle and, in Section 4, numerical results are presented. Conclusion remarks are presented in Section 5.
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Node B
MSC RNC Public switched telephone network (PSTN)
Node B
Node B
Internet protocol (IP) network
Node B
RNC
SGSN
Node B
Fig. 1. UMTS network architecture
2. Problem formulation In this section, we present the model for the expansion problem of the NSS of an UMTS wireless cellular network. 2.1. The assumptions We make the following assumptions concerning the organization of the UMTS network (see Fig. 1): (A1) each node B is connected to exactly one RNC; (A2) each RNC is connected to exactly one MSC; (A3) each RNC is connected to exactly one SGSN; (A4) each MSC is connected to the public switched telephone network (PSTN); (A5) each SGSN is connected to an Internet protocol (IP) based data network; (A6) the number of links connected to a RNC cannot exceed the maximum number of interfaces that can be installed in that RNC; (A7) the sum of the capacities of the node Bs connected to a RNC cannot exceed its switch fabric capacity (in circuit and in bps); (A8) the number of links connected to a MSC cannot exceed the maximum number of interfaces
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that can be installed in that MSC; (A9) the sum of the capacities of the links (in circuit) connected to a MSC cannot exceed its switch fabric capacity (in circuit); (A10) the number of links connected to a SGSN cannot exceed the maximum number of interfaces that can be installed in that SGSN; (A11) the sum of the capacities of the links (in bps) connected to a SGSN cannot exceed its switch fabric capacity (in bps); (A12) at most one RNC can be installed at a RNC site; (A13) at most one MSC can be installed at a MSC site; (A14) at most one SGSN can be installed at a SGSN site. We make the following assumptions concerning the expansion of the network: (A15) each link form a RNC to a MSC (or a SGSN) in the current network may be kept in place or removed from the current network; (A16) each RNC, MSC and SGSN in the current network may not be moved to another site or removed from the network; (A17) new RNCs, MSCs and SGSNs may be installed at new potential sites; (A18) the network topology is preserved in the expanded network. Finally, we suppose that the following information is known: (I1) the location of node Bs in the updated BSS; (I2) the location of facilities forming the current network; (I3) the estimate tra$c (in erlang) for each CoS between the node Bs, between the node Bs, and the PSTN and between the node Bs and the IP data network; (I4) the estimate handover tra$c (in erlang) for each CoS between the node Bs; (I5) the location of new potential RNC sites, MSC sites and SGSN sites; (I6) the di
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(in circuit) of a MSC of type s ∈ S and s the maximum number of interfaces that can be installed in a MSC of type s ∈ S to connect the RNCs); T , the set of SGSN types (where t is the switch fabric capacity (in bps) of a SGSN of type t ∈ T and t the maximum number of interfaces that can be installed in a SGSN of type t ∈ T to connect the RNCs) and C, the set of CoS (where c is the rate (in bps) of a connection of class c ∈ C). 2.2.2. Decision variables Let uij be a 0 –1 variable such that uij = 1 if and only if the node B i ∈ I is connected to a RNC installed at site j ∈ J ; vjk , a 0 –1 variable such that vjk = 1 if and only if a RNC installed at site j ∈ J is connected to a MSC installed at site k ∈ K and vjkm , the number of links of type m ∈ M connecting the RNC installed at site j ∈ J to the MSC installed at site k ∈ K. Let wjl be a 0 –1 variable such that wjl = 1 if and only if a RNC installed at site j ∈ J is connected to a SGSN installed at site l ∈ L and wjln , the number of links of type n ∈ N connecting the RNC installed at site j ∈ J to the SGSN installed at site l ∈ L. Finally, let xjr be a 0 –1 variable such that xjs = 1 if and only if a RNC of type r ∈ R is installed at RNC site j ∈ J ; yks , a 0 –1 variable such that ykt = 1 if and only if a MSC of type s ∈ S is installed at site k ∈ K and zlt , a 0 –1 variable such that zlu = 1 if and only if a SGSN of type t ∈ T is installed at site l ∈ L. 2.2.3. Tra6c variables Let fijcod (fjicod ) be the tra$c (in erlang) of class c ∈ C on the link from node B i ∈ I to site j ∈ J cod cod (fkj ), the (from site j ∈ J to node B i ∈ I ) originating from o ∈ O and destinate to d ∈ D; fjk tra$c (in erlang) of class c ∈ C on the link from site j ∈ J to site k ∈ K (from site k ∈ K to site cod j ∈ J ) originating from o ∈ O and destinate to d ∈ D and fk;cod PSTN (fPSTN; k ), the tra$c (in erlang) of class c ∈ C on the link from site k ∈ K to the PSTN (from the PSTN to site k ∈ K) originating from o ∈ O and destinate to d ∈ D. Let fjlcod (fljcod ) be the tra$c (in erlang) of class c ∈ C on the link from site j ∈ J to site l ∈ L (from site l ∈ L to site j ∈ J ) originating from o ∈ O and destinate to cod d ∈ D and fl;cod IP (fIP; l ), the tra$c (in erlang) of class c ∈ C on the link from site l ∈ L to the IP data network (from site the IP data network to site l ∈ L) originating from o ∈ O and destinate to d ∈ D. Finally, let hcod be the tra$c (in erlang) of class c ∈ C blocked at the network level originating from o ∈ O and destinate to d ∈ D. 2.2.4. Tra6c parameters Let gcod be the tra$c (in erlang) of class c ∈ C originating from o ∈ O and destinate to d ∈ D; cii d G , the handover tra$c (in erlang) from node B i ∈ I to i ∈ I for communications of class c ∈ C destinate to d ∈ D and G coii , the handover tra$c (in erlang) form node B i ∈ I to i ∈ I for communications of class c ∈ C originating from o ∈ O. 2.2.5. Cost parameters Let aqij be the link and interface costs (including the installation cost) for connecting node B i ∈ Iq to a RNC installed at site j ∈ J ; amjk , the link and interface costs (including the installation cost) for connecting a RNC installed at site j ∈ J to a MSC installed at site k ∈ K through a link and interfaces of type m ∈ M and Amjk , the cost of removing the link in the current network connecting a RNC installed at site j ∈ J to a MSC installed at site k ∈ K through a link and interfaces of type
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m ∈ M . Let anjl be the link and interface costs (including the installation cost) for connecting a RNC installed at site j ∈ J to a SGSN installed at site l ∈ L through a link and interfaces of type n ∈ N and Anjl , the cost of removing the link in the current network connecting a RNC installed at site j ∈ J to a MSC installed at site l ∈ L through a link and interfaces of type n ∈ N . Let br be the cost of a RNC of type r ∈ R (including the installation cost); cs the cost of a MSC of type s ∈ S (including the installation cost); dt , the cost of a SGSN of type t ∈ T (including the installation cost) and ec , the cost of loosing a connection of class c ∈ C in the network per erlang during one planning period (de6ned by the network planner). 2.3. The cost function In this subsection, we formulate the cost function representing the total cost of the NSS expansion process, including the cost of new facilities, the cost of installing them and removing links. In order to formulate this function, we need to represent the current network. In this paper, we identify the current network by the decision variables overlined. For instance, if node B i ∈ I is connected to the RNC installed at site j ∈ J in the current network, then uL ij = 1, and if the RNC installed at site j ∈ J is of type s ∈ S, then xLsj = 1. The cost function is composed of three terms: the cost of the links and interfaces, the cost of the nodes (i.e., the RNCs, MSCs and SGSNs) and the cost of the blocked tra$c. The cost of the links and interfaces, noted CL (u; v; w), given by the following equation, includes the cost of the new links and interfaces (including the installation cost) and the cost of removing RNC to MSC links and RNC to SGSN links from the current network. q CL (u; v; w) = aij uij + (amjk (vjkm − vLmjk )+ + Amjk (vLmjk − vjkm )+ ) q ∈ Q i ∈ Iq j ∈ J
+
j ∈ J l∈ L n ∈ N
j ∈ J k ∈ K m∈ M
(anjl (wjln − wL njl )+ + Anjl (wL njl − wjln )+ );
(1)
where (x)+ = max{0; x}. The cost of the nodes, noted CN (x; y; z), given by the following equation, includes the cost of the new RNCs, MSCs and SGSNs (including the installation cost): + s s + t t + CN (x; y; z) = xjr − xLrj + yk − yL tk + zl − zLtl : br c d (2) r ∈R
j ∈J
s ∈S
k ∈K
The cost of the blocked tra$c, noted CT (h), is CT (h) = ec hcod :
t ∈T
l∈ L
(3)
c ∈ C o∈ O d∈ D
2.4. The model The model for the expansion problem of the NSS of an UMTS wireless cellular network, noted EPN, can now be given.
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EPN: min
f;h;u;v;w;x;y;z
CL (u; v; x) + CN (x; y; z) + CT (h)
s:t:
(4)
Node B assignment constraints: uij = 1 (i ∈ I ):
(5)
j ∈J
RNC assignment constraints: vjk = xjr (j ∈ J );
(6)
r ∈R
k ∈K
wjl =
r ∈R
l∈ L
xjr
(j ∈ J ):
(7)
RNC-type uniqueness constraints: xjr 6 1 (j ∈ J ):
(8)
MSC-type uniqueness constraints: yks 6 1 (k ∈ K):
(9)
SGSN-type uniqueness constraints: zlt 6 1 (l ∈ L)
(10)
r ∈R
s ∈S
t ∈T
Node B capacity constraints: coi cid fji + fij 6 q uij c ∈C
o∈ O
c
c ∈C
o∈ O
d∈ D
fjicoi +
(i ∈ Iq ; j ∈ J; q ∈ Q);
fijcid
6 q uij
(i ∈ Iq ; j ∈ J; q ∈ Q):
i ∈ Iq
m∈ M k ∈ K
vjkm +
(12)
d∈ D
RNC capacity constraints (at the interface level): q uij 6 r1 xjr (j ∈ J ); q ∈Q
(11)
r ∈R
n ∈ N l∈ L
wjln 6
r ∈R
r2 xjr
(j ∈ J ):
(13) (14)
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RNC capacity constraints (at the switch fabric level): q uij 6 r xjr (j ∈ J ); i ∈ Iq
q ∈Q
q
uij 6
i ∈ Iq
q ∈Q
(15)
r ∈R
r ∈R
r xjr
(j ∈ J ):
(16)
MSC capacity constraints (at the interface level): vjkm 6 s yks (k ∈ K):
(17)
MSC capacity constraints (at the switch fabric level): m vjkm 6 s yks (k ∈ K):
(18)
SGSN capacity constraints (at the interface level): wjln 6 t zlt (l ∈ L):
(19)
SGSN capacity constraints (at the switch fabric level): n wjln 6 t zlt (l ∈ L):
(20)
RNC-MSC link capacity constraints: cod fjk 6 m vjkm (j ∈ J; k ∈ K);
(21)
m∈ M j ∈ J
m∈ M
s ∈S
j ∈J
s ∈S
n∈ N j ∈ J
n∈ N
t ∈T
j ∈J
t ∈T
m∈ M
c ∈ C o∈ O d∈ D
c ∈ C o∈ O d∈ D
cod fkj 6
m∈ M
m vjkm
(j ∈ J; k ∈ K);
RNC-SGSN link capacity constraints: c fjlcod 6 n wjln (j ∈ J; l ∈ L); c ∈C
c ∈C
o∈ O d ∈ D
(23)
n∈ N
o∈ O d ∈ D
c
(22)
fljcod
6
n∈ N
n wjln
(j ∈ J; l ∈ L):
(24)
CAC and CoS priority constraints: CAC and CoS priority constraints :
(25)
Tra6c =ow conservation constraints: Tra$c Jow conservation constraints :
(26)
S. Chamberland / Computers & Operations Research 31 (2004) 1769 – 1791
Additional constraints: vjk max{r2 } ¿ vjkm r ∈R
vjk 6
m∈ M
(j ∈ J; k ∈ K);
r ∈R
n∈ N
(27)
m∈ M
vjkm
wjl max{r2 } ¿ wjl 6
(j ∈ J; k ∈ K);
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n∈ N
wjln
wjln
(28)
(j ∈ J; l ∈ L);
(29)
(j ∈ J; l ∈ L):
(30)
Expansion constraints: uij ¿ uL ij xjr ¿ xLrj yks ¿ yL sk zlt ¿ zLtl
(i ∈ I; j ∈ J );
(31)
(j ∈ J; r ∈ R);
(32)
(k ∈ K; s ∈ S);
(33)
(l ∈ L; t ∈ T ):
(34)
Nonegativity constraints: 2|C OD|(|J |(|I |+|K |+|L|)+|K |+|L|)
f ∈ R+
:
(35)
Integrality constraints: vjk ∈ B; vjkm ∈ N
(j ∈ J; k ∈ K; m ∈ M );
wjl ∈ B; wjln ∈ N
(j ∈ J; l ∈ L; n ∈ N );
u ∈ B|I J | ;
x ∈ B | J R| ;
y ∈ B |K S | ;
z ∈ B | L T | :
(36)
The objective function (4) of EPN, as mentioned before, is composed of three terms, representing the expansion cost of the NSS. Constraints (5) are node B assignment constraints that require each node B to be connected to exactly one RNC and constraints (6) and (7) are RNC assignment constraints that require each RNC to be connected to exactly one MSC and exactly one SGSN. The RNC-type uniqueness constraints (8) impose that at most one RNC type be installed at site j ∈ J , MSC-type uniqueness constraints (9) impose that at most one MSC type be installed at site k ∈ K and SGSN-type uniqueness constraints (10) impose that at most one SGSN type be installed at site l ∈ L. Constraints (11) necessitate the total tra$c (in erlang) from node B i ∈ I be less or equal to its capacity (in circuit) and constraints (12) impose the total tra$c (in bps) from node B i ∈ I be less or equal to its capacity (in bps). Constraints (13) necessitate the total number of links from the node Bs to the RNC type installed at site j ∈ J be less than or equal to the maximum number of interfaces that can be installed in that RNC for the node Bs and constraints (14) impose the total number of links from the RNC type installed at site j ∈ J to the MSCs and SGSNs be less than or equal to the maximum number of interfaces that can be installed in that RNC for the MSCs and SGSNs. Constraints (15) dictate the sum of the capacities (in circuit) of the node Bs connected to the RNC type installed at site j ∈ J be less than or equal to its switch fabric capacity (in circuit)
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and constraints (16) impose the sum of the rates (in bps) of the node Bs connected to the RNC type installed at site j ∈ J be less than or equal to its switch fabric capacity (in bps). Constraints (17) necessitate the total number of links from the RNCs to the MSC type installed at site k ∈ K be less than or equal to the maximum number of interfaces that can be installed in that MSC and constraints (18) dictate the total capacity of the links (in circuit) from the RNCs to the MSC type installed at site k ∈ K be less than or equal to the switch fabric capacity (in circuit) of that MSC. Constraints (19) necessitate the total number of links from the RNCs to the SGSN type installed at site l ∈ L be less than or equal to the maximum number of interfaces that can be installed in that SGSN and constraints (20) dictate the total capacity of the links (in circuit) from the RNCs to the SGSN type installed at site l ∈ L be less than or equal to the switch fabric capacity (in circuit) of that SGSN. Constraints (21) and (22) are the RNC to MSC link capacity constraints (in circuit) and constraints (23) and (24) are the RNC to SGSN link capacity constraints (in bps). Constraints (25) detail how the connection admission is performed in the NSS and the priorities are applied between the class of services. In order to have a model that can be adapted to any standard and proprietary CAC algorithms and CoS prioritization schemes, these constraints are not speci6ed directly in the model. For instance, let C = {c0 ; : : : ; c|C |−1 } such that c0 is the class for the voice service and c1 to c|C |−1 the classes for the data services such that c1 is the 6rst priority class, c2 the second priority class, etc. As a result, ec0 ¿ · · · ¿ ec|C |−1 and constraints (25) can be formulated as follows. For the node B to RNC links, (37) fijc0 od + fjic0 od 6 q ; o∈ O d ∈ D
o∈ O d ∈ D
fijc0 od + fjic0 od ; fijc1 od + fjic1 od 6 q −
(38)
o∈ O d∈ D
··· o∈ O d ∈ D
c
(fij|C |−1
od
c
od
+ fji|C |−1 ) 6 q −
|C |−2
l=0 o∈O d∈D
(fijcl od + fjicl od );
(39)
for all i ∈ Iq , q ∈ Q and j ∈ J . Akin constraints can be written down for the other links (i.e., the RNC to MSC links and the RNC to SGSN links). Constraints (26), detailed below, are tra$c Jow conservation constraints. cod g + G ci od − hcod if o = i cod fij = (i ∈ I; c ∈ C; o ∈ O; d ∈ D); (40) i ∈I \{o} 0 otherwise j ∈J cod g + G ci od − hcod if d = i cod fji = (i ∈ I; c ∈ C; o ∈ O; d ∈ D); (41) i ∈I \{o} 0 otherwise j ∈J cod fijcod − fjk = 0 (j ∈ J; c = c0 ; o ∈ O; d ∈ D); (42) i ∈I
k ∈K
S. Chamberland / Computers & Operations Research 31 (2004) 1769 – 1791
cod fkj −
k ∈K
i ∈I
fijcod −
j ∈J
j ∈J
i ∈I
fjicod = 0
(j ∈ J; c = c0 ; o ∈ O);
(43)
fjlcod = 0
(j ∈ J; c ∈ C \ {c0 }; o ∈ O; d ∈ D);
(44)
fjicod = 0
(j ∈ J; c ∈ C \ {c0 }; o ∈ O; d ∈ D);
(45)
l∈ L
fljcod −
l∈ L
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cod fjk −
fjlcod −
i ∈I
j ∈J
j ∈J
fk;cod PSTN −
k ∈K
cod cod cod fkj + fPSTN; k − fk; PSTN = 0
cod cod fljcod + fIP; l − fl; IP = 0
k ∈K
cod fPSTN; k
cod g = −gcod 0
cod g cod cod fl; IP − fIP; l = −gcod l∈ L l∈ L 0
(k ∈ K; c = c0 ; o ∈ O; d ∈ D);
(46)
(l ∈ L; c ∈ C \ {c0 }; o ∈ O; d ∈ D):
(47)
if d = PSTN if o = PSTN
(c ∈ C; o ∈ O; d ∈ D);
(48)
otherwise
if d = IP if o = IP
(c ∈ C; o ∈ O; d ∈ D):
(49)
otherwise
Constraints (27) and (28) impose for all j ∈ J and k ∈ K, vjk = 1 if and only if m∈M vjkm ¿ 1 and constraints (29) and (30) dictate for all j ∈ J and l ∈ L, wjl = 1 if and only if n∈N wjln ¿ 1. Constraints (31) dictate the node B to RNC links installed in the current network to stay in place in the expanded network, constraints (32)–(34) impose each RNC, MSC and SGSN installed in the current network to stay in place in the expanded network and, 6nally, constraints (35) are nonnegativity constraints and constraints (36) the integrality constraints. We remind to the reader that the objective function of EPN was formulated to reach a compromise between the cost of the NSS and the cost of the blocked tra$c. However, if the network planner wants to consider a blocking probability limit at the NSS level, EPN can be easily adapted by including a new set of constraints. For instance, if a blocking probability limit of 0.1% is wanted per origin–destination, the following set of constraints can be added in the EPN formulation: c ∈C
gcod +
hcod 6 0:1 ciod + G coid ) i∈I (G
(o ∈ O; d ∈ D):
Since EPN is NP-hard (transformation from the uncapacity facility location problem), in the rest of this paper, we concentrate our e
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3. The tabu search algorithm In this section, we propose a tabu search (TS) algorithm for EPN, called TS-EPN. An introduction to tabu search can be found in [15]. The following notation is used in the presentation of TS-EPN. Let 4j be the state of the site j ∈ J such that 4j = 0 if there is no RNC installed at site j and 4j = r (for r ∈ R) if a RNC of type r is installed at site j. Similarly, let 4k be the state of the site k ∈ K such that 4k = 0 if there is no MSC installed at site k and 4k = s (for s ∈ S) if a MSC of type s is installed at site k and 6nally, let 4l be the state of the site l ∈ L such that 4l = 0 if there is no SGSN installed at site l and 4l = t (for t ∈ T ) if a SGSN of type t is installed at site l. Let y(), y() and z() be the vectors x, y and z when the state vector of all sites = {4j }j∈J {4k }k ∈K {4l }l∈L is 6xed. In the next subsection, we propose a decomposition approach to solve EPN(), i.e., the model EPN when the state vector is 6xed. 3.1. Solving EPN() When vectors x, y and z are 6xed, EPN can be decomposed into two subproblems. The 6rst subproblem, noted EPN(x; y; z), is given below. EPN(x; y; z): q aij uij min u q ∈ Q i ∈ Iq j ∈ J (50) s:t:
(5); (13); (16); (31) and u ∈ B|I J | :
(51)
The purpose of this subproblem is to connect the node Bs to the RNCs while respecting assignment, RNC degree and capacity constraints. Since this subproblem is NP-hard (transformation from the knapsack problem) and we may have to solve it several thousands instances of it during the tabu search procedure, we propose a heuristic to 6nd solutions quickly. This heuristic, called heuristic for the 6rst subproblem (HFS), is presented below. Heuristic HFS. Step 1: Order the elements in Q such that q0 ¿ q1 ¿ · · · ¿ q|Q|−1 and q0 ≺ q1 ≺ : : : ≺ q|Q|−1 . Step 2: For q := q0 to q|Q|−1 do Solve the following problem, called EPN q (u; x; y; z), that consists to connect the node Bs in Iq to the RNCs considering that the node Bs in q ∈Q:q ≺q Iq are already connected to RNCs. EPNq (u; x; y; z): q min (52) aij uij {uij :i∈Iq ; j∈J }
i ∈ Iq j ∈ J
S. Chamberland / Computers & Operations Research 31 (2004) 1769 – 1791
s:t:
uij = 1
(i ∈ Iq );
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(53)
j ∈J
i ∈ Iq
i ∈ Iq
i ∈ Iq
1 r r q uij 6 q 1 x j − uij r ∈R q ∈Q:q ≺q i∈I
(j ∈ J );
(54)
1 r r q uij 6 q xj − uij r ∈R q ∈Q:q ≺q i∈I
(j ∈ J );
(55)
1 r r q uij 6 x − u ij j q r ∈R q ∈Q:q ≺q i∈I
(j ∈ J );
(56)
q
q
q
uij ∈ R+
(i ∈ Iq ; j ∈ N ):
(57)
This subproblem is an instance of the linear assignment problem; to solve it, we use the shortest augmenting path algorithm LAPJV of Jonker and Volgenant [16]. Step 3: Return the vector u and the cost of the subproblem using (50). The second subproblem, noted EPN(u; x; y; z), is given below. EPN(u; x; y; z): min (amjk (vjkm − vLmjk )+ + Amjk (vLmjk − vjkm )+ ) f;h;v;w
j ∈ J k ∈ K m∈ M
+
j ∈ J l∈ L n ∈ N
s:t:
(anjl (wjln − wL njl )+ + Anjl (wL njl − wjln )+ ) + CT (h)
(58)
(6); (7); (11); (12); (14); (17)–(30) and 2|C OD|(|J |(|I |+|K |+|L|)+|K |+|L|)
f ∈ R+
vjk ∈ B; vjkm ∈ N
(j ∈ J; k ∈ K; m ∈ M );
wjl ∈ B; wjln ∈ N
(j ∈ J; l ∈ L; n ∈ N ):
(59)
(60)
The purpose of this subproblem is to connect the RNCs to the MSCs and SGSNs while respecting assignment constraints, RNC degree constraints, MSC degree and capacity constraints, SGSN degree and capacity constraints, link capacity constraints and tra$c Jow conservation constraints. Since this subproblem is NP-hard (transformation from the knapsack problem), we propose a heuristic to 6nd solutions quickly. This heuristic, called Heuristic for the second subproblem (HSS), is presented below.
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Heuristic HSS. Step 1: For all j ∈ J do 1.1. Calculate Tra$c (i; j; c) de6ned as the tra$c (in erlang) of class c ∈ C from node B i ∈ I to RNC j that would be switched by a MSC or a SGSN. Therefore, for all i ∈ I and c ∈ C, if uij = 1 set gcid + gcoi + G ci id Tra$c(i; j; c) := d∈ D
i ∈ I d∈ D
o∈ O
otherwise set Tra$c(i; j; c) := 0. 1.2. Calculate BlockedTra$c(i; j; c) de6ned as the tra$c (in erlang) of class c ∈ C blocked from node B i ∈ I to RNC j using constraints (25). 1.3. Calculate Tra$cMSC (j) de6ned as the tra$c (in erlang) passing through the RNC j that should be switched by a MSC and Tra$cSGSN (j), the tra$c (in bps) passing through the RNC j that should be switched by a SGSN. Therefore, (Tra$c(i; j; c0 ) − BlockedTra$c(i; j; c0 )); Tra$cMSC (j) := i∈I :uij =1
Tra$cSGSN (j) :=
c (Tra$c(i; j; c) − BlockedTra$c(i; j; c)):
i∈I :uij =1 c∈C \{c0 }
1.4. Calculate MinLinksMSC (j) de6ned as the minimum number of links needed from the RNC j to a MSC and MinLinksMSC (j), the minimum number of links needed from the RNC j to a SGSN. Therefore, Tra$cMSC (j) ; MinLinksMSC (j) := maxm∈M { m } Tra$cSGSN (j) MinLinksSGSN (j) := maxn∈N { n } and set MinLinks(j) := MinLinksMSC (j) + MinLinksSGSN (j). 1.5. If MinLinks(j) 6 r ∈R r2 xjr go to Step 1.6. Otherwise, stop, the subproblem is not feasible. 1.6. Determine by enumeration the number of links of each type NumLinksMSC (j; m) for all m ∈ M , such that the total number of links is MinLinksMSC (j) and the total capacity of these links (in erlang), noted TotalLinkCapacityMSC (j), is minimum but greater or equal to Tra$cMSC (j). 1.7. Calculate NumLinksInstallMSC (j; k; m) de6ned as the number of links of type m ∈ M to install if RNC j is connected to MSC k ∈ K and NumLinksRemoveMSC (j; k; m), the number of links of type m ∈ M to remove if RNC j is connected to MSC k ∈ K. Therefore, for all k ∈ K and m ∈ M , set NumLinksInstallMSC (j; k; m) := (NumLinksMSC (j; m) − vLmjk )+ NumLinksRemoveMSC (j; k; m) :=
k ∈K \{k }
vLmjk :
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1.8. Determine by enumeration the number of links of each type NumLinksSGSN (j; n) for all n ∈ N , such that the total number of links is MinLinksSGSN (j) and the total capacity of these links (in bps), noted TotalLinkCapacitySGSN (j), is minimum but greater or equal to Tra$cSGSN (j). 1.9. Calculate NumLinksInstallSGSN (j; l; n) de6ned as the number of links of type n ∈ N to install if RNC j is connected to SGSN l ∈ L and NumLinksRemoveSGSN (j; l; n), the number of links of type n ∈ N to remove if RNC j is connected to SGSN l ∈ L. Therefore, for all l ∈ L and n ∈ N , set NumLinksInstallSGSN (j; l; n) := (NumLinksSGSN (j; n) − wL njl )+
NumLinksRemoveSGSN (j; l; n) :=
wL njl :
l ∈L\{l}
Step 2: Let 51 ; 52 ; : : : ; 5|6| be the di
s:t:
j ∈JMSC (p) k ∈K
vjk = 1
m∈ M
(j ∈ JMSC (p));
k ∈K
1 s s vjk 6 yj − ’ p j ∈J s ∈S MSC(p)
j ∈JMSC(p)
1 vjk 6 s yjs − 5p s ∈ S
p −1 j ∈∪p =1 JMSC(p )
(62) MinLinksMSC (j)vjk
(k ∈ K);
TotalLinkCapacityMSC (j)vjk
(63)
(k ∈ K);
(64)
p −1 j ∈∪p =1 JMSC(p )
vjk ∈ R+ (j ∈ JMSC (p); k ∈ K);
(65)
where aLmjk := amjk NumLinksInstallMSC (j; k; m); AL mjk := Amjk NumLinksRemoveMSC (j; k; m): This subproblem is an instance of the linear assignment problem; to solve it, we use the shortest augmenting path algorithm LAPJV of Jonker and Volgenant [16].
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Step 4: Let 91 ; 92 ; : : : ; 9|:| be the di
s:t:
j ∈JSGSN (p) l∈L
wjl = 1
n∈ N
(j ∈ JSGSN (p));
l∈ L
j ∈JSGSN (p)
1 t t wjl 6 z − $p t ∈ T l
j ∈JSGSN (p)
1 wjl 6 t zlt − 9p t ∈ T
wjl ∈ R+
(j ∈ JSGSN (p); l ∈ L);
p −1 j ∈∪p =1 JSGSN(p )
(67) MinLinksSGSN (j)wjl (l ∈ L);
(68)
TotalLinkCapacitySGSN (j)wjl (l ∈ L);
(69)
p −1 j ∈∪p =1 JSGSN(p )
(70)
where aLnjl := anjl NumLinksInstallSGSN (j; l; n); AL njl := Anjl NumLinksRemoveSGSN (j; l; n): This subproblem is an instance of the linear assignment problem; to solve it, we use the shortest augmenting path algorithm LAPJV of Jonker and Volgenant [16]. Step 6: For all j ∈ J , k ∈ K and m ∈ M set vjkm := NumLinksMSC (j; m)vjk and for all j ∈ J , l ∈ L and n ∈ N set wjln := NumLinksSGSN (j; n)wjl . Step 7: Return the vectors v and w and the cost of the subproblem using (58). 3.2. The tabu search algorithm Each move of tabu search consists of modifying the state of a given site in the current solution (respecting constraints (32)–(34)). At each iteration of the search, we determine the best move (among the |J R| + |K S| + |L T | possible moves) while taking into account the tabus as well as the aspiration criterion. The chosen site is declared tabu for a number of iterations that is randomly determined according to a uniform discrete distribution on a speci6ed interval.
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The aspiration criterion states that if the use of tabu site allows us to discover a solution better than any other found so far, we may remove the tabu from this site. We now proceed to a detailed description of TS-EPN. Algorithm TS-EPN. Step 1: (Initial solution) Find an initial solution by solving EPN() where vector is determined as follows. • for all j ∈ J , if for a value of r ∈ R, xLrj = 1 set 4j := r, otherwise set 4j := arg maxr ∈R {r }, • for all k ∈ K, if for a value of s ∈ S, yL sk = 1 set 4k := s, otherwise set 4k := arg maxs∈S {s }, • for all l ∈ L, if for a value of t ∈ T , zLtl = 1 set 4l := t, otherwise set 4l := arg maxt ∈T {t }. Repeat Steps 2–3 for Max Iter iterations. Step 2 (Exploring the neighborhood): If the iteration count is a multiple of 10, apply the perturbation method (presented in the next subsection) to the current solution. Otherwise, do Steps 2.1 and 2.2. 2.1. Determine the best move while taking into account the tabus, the aspiration criterion and constraints (32)–(34). For each move → , which modi6es the state of a given site in the current solution, we solve EPN( ) as described in Section 3.1. The cost of a solution is given by the objective function (4). 2.2. Determine the number of iterations according to a uniform distribution on a speci6ed interval for which the chosen site is tabu. Step 3 (Best solution update): If the cost of the current solution is less than the cost of the best solution found so far, update this best solution. In the algorithm TS-EPN, the neighborhood can be fully explored or randomly explored. In the latter case, the percentage of the neighborhood explored should be speci6ed in the algorithm. Other parameters should be selected, i.e., the number of iterations (i.e., the Max Iter parameter) and the interval of the number of iteration that a site can be tabu. 3.3. The perturbation method We propose a perturbation method for visiting several local minima in few iterations of the search process. The proposed perturbation method, performed once every 10 iterations, consists of choosing the minimum cost solution obtained from the current solution by 1. the elimination of a new node (i.e., a new RNC, MSC or SGSN); 2. the installation of a new node at an empty site (i.e., a new candidate site such that its state is zero); 3. the transfer of a new node installed from a site to an empty site. Each site so that its state is changed by the perturbation method is declared tabu for a number of iterations determined in the same way as it is done in Step 2.2 of TS-EPN.
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Table 1 Features of the node B types
Capacity (circuits) Capacity (Mbps) Number of ATM interfaces
Type A
Type B
Type C
100 120 1 × OC-3
200 240 2 × OC-3
400 480 1 × OC-12
Table 2 Costs of the RNC types (including the installation costs)
Switch fabric capacity (circuits) Switch fabric capacity (Gbps) Maximum number of node B interfaces Maximum number of MSC/SGSN interfaces Cost ($)
Type A
Type B
Type C
1000 2 10 15 50,000
2500 5 20 30 90,000
5000 10 40 60 120,000
Table 3 Costs of the MSC types (including the installation costs)
Switch fabric capacity (circuits) Maximum number of interfaces Cost ($)
Type A
Type B
Type C
100,000 50 200,000
200,000 100 350,000
300,000 150 500,000
Type A
Type B
Type C
20 16 40,000
40 32 60,000
80 64 80,000
Table 4 Costs of the SGSN types (including the installation costs)
Switch fabric capacity (Gbps) Maximum number of interfaces Cost($)
4. Numerical results In this section, we present a study of the performance of the proposed heuristics. The algorithm TS-EPN was implemented in the C language on a SunFire 4800 workstation (4 CPU, 900 MHz and 4 GB of RAM). For the tests, three node B types, three RNC types, three MSC types and three SGSN types are used. Their features are presented respectively in Tables 1–4. Moreover, asynchronous transfer mode (ATM) OC-3 and OC-12 links are used to connect the node Bs to the RNCs, DS-3 links are used to connect the RNCs to the MSCs and gigabit Ethernet (GE) links to
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Table 5 Costs of the links (including the installation costs) Link type
Capacity
Cost ($/km)
DS-3 OC-3 OC-12 GE
2688 circuits 149:760 Mbps 594:432 Mbps 1 Gbps
2500 1500 4000 4000
Table 6 Costs of the interface types (including the installation costs) Interface type
Cost ($)
DS-3 OC-3 OC-12 GE
1500 2000 4500 2000
connect the RNCs to the SGSNs. The link costs are presented in Table 5 and the interface costs in Table 6. Finally, the cost of removing a link is 100$=km. Three classes of service are considered for the tests. The 6rst class (class 1) is used for voice service (using a 16 kbps compression), the class 2 is used for the 144 kbps data services and, the class 3 for the 384 kbps data service. The cost of loosing a connection of class 1, 2 and 3 during per erlang per planning period is 180$, 200$ and 220$ respectively. For the tests, the Max Iter parameter of the TS-EPN heuristic is set to 100 and the interval of the number of iteration that a site can be tabu to [5,10] because the best solutions was found within 100 iterations and with this interval. Moreover, at each iteration of TS-EPN, the neighborhood is fully explored. TS-EPN results were compared to a lower bound (denoted LB) obtained as follows. The bound LB was obtained by using the CPLEX Mixed Integer Solver [17] to solve EPN without constraints (11), (12), (21)–(26) and integrality constraints on v and w variables. The default settings of CPLEX are used and the branch-and-bound node limit was set to 100,000 and the upper limit on the size of the tree to 100 MB. For the evaluation of the heuristic performance for network design, 40 design problems are generated randomly as follows. |I | points corresponding to node Bs’ locations, |J | points corresponding to the potential RNC sites, |K| points corresponding to the potential MSC sites and |L| points corresponding to the potential SGSN sites were generated in a square region of length of 100 km following a uniform distribution. The type of each node B is selected randomly among the three node B types considered (see Table 1). The call tra$c between each pair of node Bs and between the node Bs and the PSTN and IP data network is generated randomly in the interval [0; 5] erlang,
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following a uniform distribution, for each class of service. Finally, the hanover tra$c between each pair of node Bs is generated randomly in the interval [0; 1] erlang, following a uniform distribution, for each destination and class of service. The results for the design problems are presented in Table 7 . In this table, columns 1– 4 present respectively the number of node Bs, the number of potential RNC sites, the number of potential MSC sites and the number of potential SGSN sites. Columns 5 –7 present the solution found by algorithm TS-EPN, the CPU execution time to 6nd it and the GAP that indicates the percentage gap between the heuristic solution and the lower bound LB (with respect to the value of the lower bound). The following observations can be made from Table 7. First, the heuristic TS-EPN 6nds solutions close to optimality for the design problems considered (within 2.65%, on average, from the lower bound) and the mean CPU time is 758 s. Moreover, the reader should point out that the heuristic 6nds the optimal solution four times (i.e., on average, TS-EPN 6nd the optimal solution 10% of the time for the design problems considered). For the evaluation of the heuristic performance for network expansion, 40 expansion problems are generated randomly. The results are presented in Table 8. In this table, columns 1 to 4 present respectively the number of node Bs, the number RNCs, the number MSCs and the number SGSNs in the initial network. Columns 5 –8 present respectively the number of new node Bs, the number of new potential RNC sites, the number of new potential MSC sites and the number of new potential SGSN sites. Columns 9 –11 present the solution found by algorithm TS-EPN, the CPU execution time to 6nd it and the GAP that indicates the percentage gap between the heuristic solution and the lower bound LB (with respect to the value of the lower bound). The following observations can be made from Table 8. First, the heuristic TS-EPN 6nds solutions close to optimality for the expansion problems considered (within 8.07%, on average, from the lower bound) and the mean CPU time is 2818 s. Moreover, the heuristic 6nds the optimal solution three times. For all solutions found having a gap greater than 2%, more than one link is installed between at least one RNC to a MSC or a SGSN. However, since the Jow variables and constraints are not considered to 6nd the value of LB, a unique link is used from each RNC to the MSCs and a unique link is used from each RNC to the SGSNs. We have tried to improve the value of LB by considering the Jow variables and constraints in the LB formulation but we did not succeed to solve it using CPLEX. 5. Conclusions In this paper we have studied the problem of expanding UMTS cellular wireless networks and proposed an optimization model for the location of new RNCs, MSCs and SGSNs, selecting their types, updating the network connectivity, while considering the call and handover blocking. After de6ning network elements of such networks and making some realistic assumptions about their organization, we have presented a mathematical formulation of this planning problem. A tabu search algorithm was proposed to 6nd good solutions. All the results were compared to a proposed lower bound obtained by relaxing a subset of the original constraints in the model. These results have shown that the tabu based heuristic produces
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Table 7 Computational results for the design problems Number of node Bs
Number of potential RNC sites
Number of potential MSC sites
Number of potential SGSN sites
TS-EPN OBJ
CPU
GAP
20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100
5 5 5 5 5 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20 5 5 5 5 5 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
2050.3 4134.3 6053.0 9063.4 10026.4 2257.5 3863.9 5663.1 6879.1 7290.8 2295.7 3971.3 5651.3 6064.7 6669.7 2180.1 3170.4 4532.2 5594.8 7246.4 2134.5 3985.8 7014.2 8082.3 8438.6 2180.0 3862.5 4454.5 6318.6 7790.4 2257.2 3536.9 4792.3 5761.2 7169.6 1788.9 3400.1 4721.2 6111.7 6977.6
10 40 156 498 1006 18 60 225 756 1465 35 88 302 980 1907 69 136 416 1258 2428 41 88 281 933 1814 70 130 382 1192 2257 120 197 516 1450 2755 199 288 677 1790 3301
0.00 2.22 2.17 7.51 3.77 2.26 0.88 1.56 2.94 5.45 0.00 2.06 2.63 3.77 5.03 2.78 0.65 1.65 2.36 5.07 0.19 0.98 3.64 5.95 7.16 0.00 0.54 1.68 5.96 4.76 0.83 0.00 0.59 3.63 3.50 0.97 0.91 3.61 2.96 3.50
5135.9
758
2.65
Mean
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Table 8 Computational results for the expansion problems Initial network
New node Bs and sites
TS-EPN
Number of node Bs
Number of RNCs
Number of MSCs
Number of SGSNs
Number of new node Bs
Number of new potential RNC sites
Number of new potential MSC sites
Number of new potential SGSN sites
OBJ (k$)
CPU (s)
GAP (%)
20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 80 80 80 80 80 80 80 80 100 100 100 100 100 100 100 100
2 4 4 4 3 4 3 3 5 5 6 5 3 5 6 6 4 5 7 8 3 5 6 8 4 8 8 8 4 6 8 10 5 8 10 10 5 8 9 10
1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 1 1 1 2 2 2 1 3 2 1 2 2 1 3 2 2 3 2 3 3 2 2
1 2 2 2 2 3 2 2 3 2 2 3 2 3 3 3 1 2 2 2 2 4 4 4 2 4 4 3 2 3 5 4 2 2 3 3 3 4 4 5
20 20 40 40 60 60 80 80 20 20 40 40 60 60 80 80 20 20 40 40 60 60 80 80 20 20 40 40 60 60 80 80 20 20 40 40 60 60 80 80
5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10
5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10
5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10
1905.1 1923.7 4150.1 2855.4 7055.1 4308.2 6943.2 5518.8 1830.7 1997.8 3744.9 2815.4 7319.8 4306.0 6670.7 5383.6 1432.1 1729.7 3311.7 2908.7 7686.0 4450.8 6753.2 4714.3 1810.8 1498.3 3587.2 2777.8 6056.0 3627.8 5121.3 4513.0 1328.3 1458.6 3096.8 2585.8 7007.3 4174.0 5637.7 4553.9
41 151 154 402 525 1194 1030 2252 155 397 539 1179 1041 2244 1764 3769 539 1182 1060 2266 1778 3803 2808 5944 1066 2233 1825 3809 2837 5940 4245 8744 1806 3693 2850 5921 4181 8791 5984 12563
0.00 0.00 2.55 0.73 5.22 4.80 10.20 3.50 12.30 7.97 12.63 7.44 6.99 6.32 7.90 2.47 6.87 16.93 16.61 9.11 5.61 4.65 13.41 8.57 0.00 8.20 11.38 10.52 3.64 9.93 12.16 5.65 6.43 23.67 7.26 24.70 1.83 3.94 11.00 9.71
4013.7
2818
8.07
Mean
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solutions close to the lower bound for test networks. For the design test problems, the heuristic has found solutions within, on average, 2.65% from the lower bound and for the expansion problem, the heuristic has found solutions within, on average, 8.07% from the lower bound. As a result, the heuristics proposed are recommended for large-size cellular networks with an important number of cells. It would be interesting to improve the lower bound formulation in order to evaluate more e