2003 Kluwer Academic Publishers. Manufactured in The Netherlands.
Channel Allocation for GPRS with Buffering Mechanisms
PHONE LIN
Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan 106, ROC
Abstract. General Packet Radio Service (GPRS) provides mobile users end-to-end packet-switched services by sharing the radio channels with voice and circuit-switched services. In such a system, radio resource allocation for circuit-switched and packet-switched services is an important issue, which may affect the QoS for both services significantly. In this paper, we propose two algorithms: Dynamic Resource Allocation with Voice and Packet queues (DRAVP) and Dynamic Resource Allocation with Packet and Voice queues (DRAPV) for channel allocation of the voice calls and packets. We propose analytic and simulation models to investigate the performance of DRAVP and DRAPV in terms of voice call incompletion probability, packet dropping probability, average voice call waiting time, and average packet waiting time. Our study indicates that the buffering mechanism for GPRS packets significantly increase the acceptance rate of GPRS packets at the cost of slightly degrading the performance of voice calls.
Keywords: GPRS, GSM, buffering mechanism, dynamic channel allocation
1. Introduction
General Packet Radio Service (GPRS) [6,7] is an end-to-end
packet-switched protocol to provide mobile users applications such as the World Wide Web (WWW), where users spend most of time reading information, and the bursty data are trans-ferred through the link only when necessary. GPRS is consid-ered as a bearer service for mobile networks (e.g., GSM [14], IS-136 [9], or UMTS [2]), which greatly improves and sim-plifies the wireless access to packet data networks (e.g., the Internet or X.25). Most cellular operators reuse the exist-ing GSM infrastructure to provide the GPRS service. In this paper, we assume that the mobile network for GPRS is GSM. Compared with the previous mobile data services (e.g., circuit-switched data and short message services), users of GPRS benefit from shorter access times and higher data rates. Figure 1 illustrates the GPRS/GSM architecture. In the architecture, the Base Station System (BSS) consists of one
Base Station Controller (BSC) and several Base Transceiver Stations (BTSs). The BSC is connected to the Serving GPRS Support Node (SGSN) and Mobile Switching Center (MSC)
for the provisions of packet-switched (i.e., IP) and circuit-switched (i.e., PSTN) services, respectively. The Mobile
Sta-tion (MS) communicates with a BTS through the radio
in-terface Um [8] based on the TDMA technology, where the radio coverage of a BTS is referred to as cell. The SGSN is responsible for delivery the packets to the MS, and the
Gate-way GPRS Support Node (GGSN) acts as a gateGate-way between
GPRS and the external data networks. The existing GSM net-work nodes including BSS, Mobile Switching Center/Visitor
Location Register (MSC/VLR), and Home Location Register
(HLR) are upgraded to accommodate GPRS.
The GPRS air interface [7] has been implemented for com-munication between the MS and BSS, which shares the phys-ical channels with the voice calls and circuit-switched ser-vices. The operator may dynamically allocate the physical channels for both voice calls and packet usage. The physical
channel dedicated to packet data traffic is called a Packet Data Channel (PDCH). Three types of packet data logical channels are defined in GPRS: Packet Data Traffic Channel (PDTCH),
Packet Common Control Channel (PCCCH) and Packet Ded-icated Control Channel (PDCCH). These different types of
logical channels can camp on the same PDCH. The PDTCH, PCCCH and PDCCH are used for user packet data transfer, the GPRS common control signaling delivery, and the dedi-cated control signaling delivery for an MS, respectively. Al-location of channels for a GPRS user is flexible where one to eight channels can be allocated to a user, or one channel can be shared by several users. To initiate the uplink packet transfer, the MS executes the following steps.
Step 1. An MS negotiates with the network for the radio
re-source via PCCCH or PDCCH.
Step 2. According the agreed resource assignment from the
network, the MS starts to transmit packets to the network.
Step 3. During the packet transmission, if the MS requires
more PDTCHs, it can specify the request through an as-signed uplink radio block.
Step 4. The network and the MS then exchange resource
re-assignment messages through the PDCCHs to re-allocate the resources for uplink transmission. The MS continues to transmit packets.
Step 5. When the MS completes the transmission, the MS
and network exchange the final data block indication and the final data block acknowledgement.
The downlink packet transfer is similar to the uplink packet transfer, and is not described here. Details of the packet transfer procedure can be found in [8]. Note that in this procedure, the amount of radio resource (i.e., number of PDCHs) for a packet request will be recorded in the QoS pro-file of the user at the SGSN. In resource assignment (step 1) and re-assignment (step 4), the BSS may “dynamically” or
Figure 1. The GPRS/GSM architecture.
“statically” allocate radio channels to the MS. In static chan-nel allocation, the requested amount of PDCHs is allocated for the packet request. On the other hand, in dynamic channel allocation, the BSS allocates partial amount of PDCHs for the packet request. If no radio channel is available when step 1 is executed, the packet request is buffered in the MS (for uplink packet transfer) or BSS (for downlink packet transfer) [1].
Since the GPRS packet service shares the radio channels with the GSM voice service, how to efficiently allocate ra-dio channels for both GSM voice calls and GPRS packet requests is an important issue, which may affect the QoS for both GSM voice and GPRS packet services significantly. In [12], we proposed four channel allocation algorithms for the GPRS packets and GSM voice calls. This study indicates that the dynamic allocation for packet transmission and wait-ing queue for voice calls (new calls or handoff calls) signif-icantly improve the performance of the network. However, the buffering mechanism for GPRS packets is not addressed in [12]. In this paper, we propose two channel allocation algorithms: Dynamic Resource Allocation with Voice and Packet queues (DRAVP) and Dynamic Resource Allocation with Packet and Voice queues (DRAPV) for scheduling of the radio channels for GPRS packets and GSM voice calls, where a voice queue (VQ) and packet queue (PQ) are used to buffer the voice calls and packet requests that are not served immediately. In DRAVP, the buffered voice calls have higher priority over the buffered packets. In DRAPV, the buffered packets have higher priority to be served than the buffered voice calls. The analytic models and simulation experiments are used to investigate the performance of the proposed algo-rithms. Our study indicates that the buffering mechanism for GPRS packet requests significantly reduces the packet drop-ping probability by affecting the performance of the voice calls slightly.
2. Dynamic channel allocation algorithms with voice and packet queues
This section describes two dynamic channel allocation al-gorithms DRAVP and DRAPV for both GPRS packets and
GSM voice calls (either a new voice call request or a hand-off voice call). Every GPRS packet request specifies the re-quested QoS profile for the number of channels required for transmission. At negotiation, the BSS allocates one or more channels for this packet request based on the negotiated QoS profile. In DRAVP and DRAPV, the BSS dynamically allo-cates the channels to a packet request based on the number of free channels in a cell. Two First-In-First-Out (FIFO) queues, VQ and PQ, are maintained in the BSS to buffer the voice call (new call or handoff call) and packet requests that are not served immediately due to that there is no free channel. When there are free channels at the cell, the requests in VQ and PQ are served based on a priority order. In GSM, the same chan-nel assignment procedure is used for both the new voice calls and handoff voice calls. This non-prioritized scheme (i.e., VQ is used to buffer both new calls and handoff calls) is con-sidered in this paper. In our previous study [12], the prior-ity scheme for new call and handoff call has been addressed. Suppose that there are L free channels at a cell when a request (either a GPRS packet or GSM voice call) arrives. The details of DRAVP and DRAPV are described as follows.
Algorithm DRAVP. For a data session that requests K
chan-nels, the BSS dynamically assigns channels as follows. If
L K, the BSS assigns K channels to the packet request.
If 0 < L < K, then L channels are allocated to the re-quest. If L = 0, this request is buffered into PQ. For a voice call request, if L > 0, the BSS assigns one chan-nel to it. Otherwise (i.e., L = 0), the voice call request is buffered into the VQ. When free channels are available, the voice call requests in VQ are served immediately. If VQ is empty, then the BSS dynamically allocates channels to the packet requests in the PQ.
Algorithm DRAPV. This algorithm is similar to DRAVP
ex-cept that the requests in the PQ have higher priority to be served than that in the VQ. That is, when free channels are available, the BSS first dynamically assigns channels to the buffered packet requests, and then to the buffered voice call requests.
3. Models for algorithms DRAVP and DRAPV
In this paper, we develop simulation models for the DRAVP and DRAPV algorithms, respectively. Analytic models are constructed to validate the simulation experiments. The sim-ulation models follow the discrete event simsim-ulation approach in [12], and a 6× 6 wrapped mesh cell structure is considered in our experiments. The input parameters set up and output measures evaluated in our study are listed in appendix.
In our analytic models, we assume that the GSM voice call arrivals and GPRS packet requests to a cell form Pois-son streams with rates λvand λp, respectively. Let tcvbe the voice call holding time, which is assumed to be exponentially distributed with the density function fcv(tcv)= µve−µvtcvand the mean voice call holding time E[tcv] = 1/µv. Let tcpbe the packet transmission times. If one channel (k channels) is (are) allocated to the packet, then the density function for the packet transmission times is fcp(tcp)= µpe−µptcp(fcp(tcp)=
kµpe−kµptcp) with mean E[tcp] = 1/µp(E[tcp] = 1/(kµp)). Note that in the real world, the packet inter-arrival times and packet transmission times may not be exponential distribu-tion. By using exponential assumptions, our analytic models served for two purposes. First, exponential distribution pro-vides the mean value analysis. Second, the analytic models are for validation of the simulation experiments that we use to investigate the performance of DRAVP and DRAPV. In the GSM/GPRS network, if an MS moves to another cell during the conversation, then the radio link to the old cell is discon-nected, and a radio link to the new cell is required to continue the conversation. This process is called handoff [5]. If the new cell does not have any idle channel, the handoff call is
forced to terminate. In our study, we consider the mobility
of voice users but ignore the effects of mobility (handoff) on the GPRS packet transmission. This assumption is justified as follows. Although a GPRS session can be elapsed for a long period, the individual packet transmission times are short, and the handoff procedure can be initiated after the current packet transmission is completed. On the other hand, voice call hold-ing times are long enough so that handoffs may occur durhold-ing the conversation. Thus the handoff effects of voice calls must be considered.
3.1. The analytic model DRAVP0
This section describes the analytic model DRAVP0, where we consider the packet traffic by ignoring voice call arrivals to a cell. We use a (K+ 1)-state Markov process to derive the packet dropping probability Pbp. A state (nPQ, npK, npK−1, npK−2, . . . , np1)denotes that in a cell, nPQpacket requests are
buffered in the PQ, and npK packets (each allocated K
chan-nels), npK−1packets (each allocated K − 1 channels), npK−2
packets (each allocated K− 2 channels), . . . , and np1 packets
(each allocated one channel) are being served. For the illustra-tion purpose, we consider K = 3 in our discussion. Suppose that there are C channels in a cell. The maximum number of packet requests that can be buffered in the PQ is P . In this Markov process, a state is represented by (i, j, k, l) where
Figure 2. The state transition diagram for DRAVP0.
i = nPQ, j = np3, k = np2, l = np1. The state space S1for
this Markov process is
S1= (i, j, k, l)i= 0, 0 3j + 2k + l C, 0 j C 3 , 0 k C 2 and 0 l C ∪ (i, j, k, l)0 < i P, 3j + 2k + l = C, 0 j C 3 , 0 k C 2 and 0 l C .
Let πi,j,k,l be the steady state probability for state (i, j, k,
l), where πi,j,k,l = 0 if state (i, j, k, l) /∈ S1. For all legal states (i, j, k, l) ∈ S1,
(i,j,k,l)∈S1πi,j,k,l = 1. Figure 2
illustrates the transition diagram for this Markov process. In this figure we consider the following transitions for state
(i, j, k, l)∈ S1.
• If a GPRS request arrives at state (i, j, k, l) ∈ S1where there is no free channel, and PQ is not full (i.e., 3j + 2k + l = C and 0 i < P ), then this request is buffered in the PQ. Therefore the transition from states (i, j, k, l) to (i+ 1, j, k, l) occurs only when 3j + 2k + l = C and 0 i < P . Define δ1as
δ1=
1, if 0 i < P and 3j + 2k + l = C,
0, otherwise. (1)
The process moves from state (i, j, k, l) to (i+ 1, j, k, l) with rate λpδ1.
• If the transmission for a GPRS packet (which may be al-located three, two, or one channels) completes at state
(i+ 1, j, k, l) ∈ S1, then one GPRS packet request in PQ will be served. Define δ2as
δ2=
1, if (i+ 1, j, k, l) ∈ S1,
0, otherwise. (2)
Then the process moves from state (i + 1, j, k, l) to
• If a GPRS request arrives at state (i, j, k, l) ∈ S1, where 3j+ 2k + l C − 3, then three channels are allocated to it. Define δ3as
δ3=
1, if 3j+ 2k + l C − 3,
0, otherwise. (3)
The process moves from state (i, j, k, l) to (i, j + 1, k, l) with rate λpδ3.
• When the transmission for a GPRS packet allocated three channels completes at state (i, j + 1, k, l) ∈ S1, and no GPRS packet requests are buffered in the PQ, three channels will be released. The process moves from state
(i, j+ 1, k, l) to (i, j, k, l) with rate 3(j + 1)µpδ4, where
δ4=
1, if i= 0 and (i, j + 1, k, l) ∈ S1,
0, otherwise. (4)
• When a GPRS request arrives at state (i, j, k, l) ∈ S1, where 3j+ 2k + l = C − 2, then this request is allocated two channels. Define δ5as
δ5=
1, if 3j+ 2k + l = C − 2,
0, otherwise. (5)
Then the process moves from state (i, j, k, l) to (i, j, k+ 1, l) with rate λpδ5.
• When the transmission for a GPRS packet allocated two channels completes at state (i, j, k+ 1, l), and no GPRS packet requests are buffered in the PQ, two channels will be released. The process moves from state (i, j, k+ 1, l) to (i, j, k, l) with rate 2(k+ 1)µpδ6, where
δ6=
1, if i= 0 and (i, j, k + 1, l) ∈ S1,
0, otherwise. (6)
• If a GPRS request arrives at state (i, j, k, l) ∈ S1, where 3j+ 2k + l = C − 1, then one channel is assigned to this request. Define δ7as
δ7=
1, if 3j+ 2k + l = C − 1,
0, otherwise. (7)
The process moves from state (i, j, k, l) to (i, j, k, l+ 1) with rate λpδ7.
• If the transmission for a GPRS packet allocated one chan-nel completes at state (i, j, k, l+ 1), and no GPRS packet requests are buffered in the PQ, the process moves from state (i, j, k, l+ 1) to (i, j, k, l) with rate µpδ8, where
δ8=
1, if i= 0 and (i, j, k, l + 1) ∈ S1,
0, otherwise. (8)
The transitions between (i, j, k, l) and (i − 1, j, k, l),
(i, j − 1, k, l), (i, j, k − 1, l), (i, j, k, l − 1) are similar to
that between (i, j, k, l) and (i + 1, j, k, l), (i, j + 1, k, l),
(i, j, k+ 1, l), (i, j, k, l + 1). The balance equations for this
process is: (δ1+ δ3+ δ5+ δ7)λp+ δ9(3j+ 2k + l) + δ113j + δ132k+ δ15l µp πi,j,k,l = δ2(3j+ 2k + l)µpπi+1,j,k,l+ δ43(j+ 1)µpπi,j+1,k,l + δ62(k+ 1)µpπi,j,k+1,l+ δ8(l+ 1)µpπi,j,k,l+1 + δ10λpπi−1,j,k,l+ δ12λpπi,j−1,k,l + δ14λpπi,j,k−1,l+ δ16λpπi,j,k,l−1, (9) where δ1, δ2, δ3, . . . , δ16 are obtained from (1), (2), (3), . . . , (17), respectively, and δ9= 1, if i > 0 and 3j+ 2k + l = C, 0, otherwise; (10) δ10= 1, if 3j+ 2k + l = C and (i− 1, j, k, l) ∈ S1, 0, otherwise; (11) δ11= 1, if i= 0 and (i, j − 1, k, l) ∈ S1, 0, otherwise; (12) δ12= 1, if i= 0, 3(j − 1) + k + l C − 3 and (i, j − 1, k, l) ∈ S1, 0, otherwise; (13) δ13= 1, if i= 0 and (i, j, k − 1, l) ∈ S1, 0, otherwise; (14) δ14= 1, if i= 0, 3j + 2(k − 1) + l = C − 2 and (i, j, k− 1, l) ∈ S1, 0, otherwise; (15) δ15= 1, if i= 0 and (i, j, k, l − 1) ∈ S1, 0, otherwise; (16) δ16= 1, if i= 0, 3j + 2k + (l − 1) = C − 1 and (i, j, k, l− 1) ∈ S1, 0, otherwise. (17)
When a packet request arrives at the states where the PQ is full, and there is no free channel (i.e., i = P and 3j +2k+l =
C), this request will be dropped. Therefore
Pbp =
(i,j,k,l)∈{(a,b,c,d) | a=P, 3b+2c+d=C,
(a,b,c,d)∈S1}
πi,j,k,l. (18)
From (9) and (18), the steady state probabilities πi,j,k,l and
Pbpcan be computed by using the iterative algorithm in [13]. 3.2. The analytic model DRAVP1
This section proposes the analytic model DRAVP1 for the DRAVP algorithm when K = 1 (i.e., the number of channels specified in the GPRS packet request is one). In the model, the cell residence times for a GSM voice user are assumed to have exponential distribution with mean 1/ηvand Laplace transform
fm∗(s)= ηv ηv+ s
. (19)
We use the handoff traffic model in [13] to derive handoff traffic for the voice calls, and then obtain the voice call in-completion probability Pncv and packet data dropping
prob-ability Pbp by using a Markov process. In the DRAVP
al-gorithm, the channel assignments for the handoff voice calls and the new voice calls are not distinguishable. Thus the new
voice call blocking probability Pbvand the handoff call
force-termination probability Pfv are the same, that is,
Pbv = Pfv. (20)
Let λvhbe the voice handoff call arrival rate to a cell, and
Pncv be the voice call incompletion probability. From [13]
and (19), λvhand Pncv are expressed as λvh= η2v(1− Pbv)λv µv(ηv+ µvPbv) (21) and Pncv = Pbv+ ηv2(1− Pbv) µv(ηv+ µvPbv) Pfv. (22)
Suppose that the size of the VQ and PQ are V and P , respec-tively. We model this problem by a four-dimensional Markov process. A state in this process is defined as (m, n, o , p) where m is the number of the buffered voice calls, n is the number of the voice calls being served, o is the number of the buffered packets, and p is the number of the packets (each al-located one channel) being served in the cell. The state space
S2for this Markov process is S2= (m, n, o , p)| 0 m V, 0 o P and n + p = C ∪(m, n, o , p)| m = 0, o= 0 and 0 n + p < C.
Let πm,n,o,p∗ denote the steady state probability for state
(m, n, o , p), where πm,n,o,p∗ = 0 if state (m, n, o , p) /∈ S2. For all legal states (m, n, o , p) ∈ S2,(m,n,o,p)∈S2πm,n,o,p∗ = 1. Let "v = λv+ λvh be the net new and handoff voice call arrival rate to a cell. Let 1/Mv = 1/(µv+ ηv)be the mean channel occupancy time of a voice call in a cell. The transition diagram for this process is shown in figure 3. For state (m, n, o , p) ∈ S2, we consider the state transitions for the DRAVP1 in three cases.
Case 1. We consider the transitions between states (m, n,
o , p)and (m− 1, n, o , p), (m, n − 1, o , p), (m, n, o − 1, p),
(m, n, o , p− 1).
• If a new voice call or handoff call arrives at state (m − 1, n, o , p)∈ S2where n+ p = C and 0 < m V , this voice call request will be queued in the VQ. Define δ∗1as
δ∗1= 1, if n+ p = C, 0 < m V and (m− 1, n, o , p) ∈ S2, 0, otherwise. (23) Then the process transits from state (m − 1, n, o , p) to
(m, n, o , p)with rate "vδ∗1.
• At state (m, n, o , p) ∈ S2where n+ p = C and 0 < m V , one of the n voice calls being served may complete or handoff to another cell (i.e., one of them may release the channel with rate Mv), and one of the m buffered voice calls may leave the cell with rate ηv before it is served.
(a)
(b)
(c)
Figure 3. The state transition diagram for DRAVP1. (a) Case 1. (b) Case 2. (c) Case 3.
Therefore the process moves from state (m, n, o , p) to
(m− 1, n, o , p) with rate (nMv+ mηv)δ2∗where
δ2∗= 1, if n+ p = C and (m− 1, n, o , p) ∈ S2, 0, otherwise. (24)
• When a new voice call or handoff call arrives at state
(m, n− 1, o , p) ∈ S2where n+ p − 1 < C, one channel is allocated to this voice call request. Define δ∗3as
δ3∗= 1, if n+ p − 1 < C and (m, n− 1, o , p) ∈ S2, 0, otherwise. (25)
The process moves from state (m, n− 1, o , p) to (m, n,
o , p)with rate "vδ3∗.
• A served voice call releases the channel at state (m, n, o , p) ∈ S2where m= 0 and o = 0 (i.e., no voice call or packet
requests are buffered in the VQ or PQ). Define δ4∗as δ∗4= 1, if m= 0, o = 0 and (m, n− 1, o , p) ∈ S2, 0, otherwise. (26)
The process moves from state (m, n, o , p) to (m, n − 1, o , p) with rate nMvδ4∗.
• When a GPRS packet request arrives at state (m, n, o − 1, p) ∈ S2 where n + p = C, the packet request is buffered into PQ. The process moves from (m, n, o−1, p) to (m, n, o , p) with rate λpδ∗5where
δ∗5= 1, if n+ p = C and (m, n, o− 1, p) ∈ S2, 0, otherwise. (27) • If the transmission for a GPRS packet completes at state
(m, n, o , p) ∈ S2where p 1, m = 0, and 0 < o P , one buffered GPRS packet request will be served. Define
δ∗6as δ∗6= 1, if m= 0, 0 < o P , p 1 and (m, n, o− 1, p) ∈ S2, 0, otherwise. (28)
The process moves from state (m, n, o , p) to (m, n, o− 1, p) with rate pµpδ6∗.
• If a GPRS packet request arrives at state (m, n, o , p −1) ∈
S2where n+ p − 1 < C, then this request is allocated one channel. Define δ7∗as δ∗7= 1, if n+ p − 1 < C and (m, n, o , p− 1) ∈ S2, 0, otherwise. (29)
The process moves from state (m, n, o , p− 1) to (m, n,
o , p)with rate λpδ∗7.
• When the transmission of a GPRS packet completes at state (m, n, o , p) where m = 0, o = 0, one channel is released, and the process moves from state (m, n, o , p) to
(m, n, o , p− 1) with rate pµpδ∗8where
δ∗8= 1, if m= 0, o = 0 and (m, n, o , p− 1) ∈ S2, 0, otherwise. (30)
Case 2. In this case, we consider the transitions between states (m, n, o , p) and (m + 1, n, o , p), (m, n + 1, o , p),
(m, n, o , p+1), which are similar to that between (m, n, o , p)
and (m − 1, n, o , p), (m, n − 1, o , p), (m, n, o − 1, p), (m, n, o , p− 1), and δ9∗= 1, if n+ p < C and (m+ 1, n, o , p) ∈ S2, 0, otherwise; (31) δ10∗ = 1, if n+ p = C and (m+ 1, n, o , p) ∈ S2, 0, otherwise; (32) δ11∗ = 1, if n+ p < C and (m, n+ 1, o , p) ∈ S2, 0, otherwise; (33) δ12∗ = 1, if m= 0, o = 0 and (m, n+ 1, o , p) ∈ S2, 0, otherwise; (34) δ13∗ = 1, if n+ p = C and (m, n, o+ 1, p) ∈ S2, 0, otherwise; (35) δ14∗ = 1, if m= 0 and (m, n, o+ 1, p) ∈ S2, 0, otherwise; (36) δ15∗ = 1, if n+ p < C and (m, n, o , p+ 1) ∈ S2, 0, otherwise; (37) δ16∗ = 1, if n+ p = C and (m, n, o , p+ 1) ∈ S2, 0, otherwise. (38)
Case 3. This case considers the transitions between state
(m, n, o , p)and (m−1, n+1, o , p−1), (m, n−1, o−1, p+1),
(m+ 1, n − 1, o , p + 1), (m, n + 1, o + 1, p − 1).
• If the transmission for the GPRS packet completes at state
(m, n, o , p) ∈ S2where m > 0, then the released chan-nel will be allocated to one buffered voice call request. The process moves from state (m, n, o , p) to (m− 1, n + 1, o , p− 1) with rate pµpδ∗17where
δ∗17= 1, if m > 0 and (m− 1, n + 1, o , p − 1) ∈ S2, 0, otherwise. (39)
• If a served voice call either completes or hands off to an-other cell at state (m, n, o , p) ∈ S2 where m = 0 and
o >0, then one buffered GPRS packet request will be al-located one channel. Thus the process moves from state
(m, n, o , p)to (m, n− 1, o − 1, p + 1) with rate nMvδ18∗ where δ∗18= 1, if m= 0, o > 0 and (m, n− 1, o − 1, p + 1) ∈ S2, 0, otherwise. (40)
• When the transmission for a GPRS packet completes at state (m+ 1, n − 1, o , p + 1) ∈ S2, the BSS will allocate one channel to one buffered voice call. Define δ∗19as
δ∗19=
1, if (m+ 1, n − 1, o , p + 1) ∈ S2,
0, otherwise. (41)
The process moves from state (m+ 1, n − 1, o , p + 1) to
(m, n, o , p)with rate (p+ 1)µpδ19∗.
• When a served voice call either completes or hands off to another cell at state (m, n+ 1, o + 1, p − 1) ∈ S2where
m= 0, then the released channel will be allocated to one
buffered packet request. Define δ20∗ as
δ20∗ = 1, if m= 0 and (m, n+ 1, o + 1, p − 1) ∈ S2, 0, otherwise. (42)
The process moves from state (m, n+ 1, o + 1, p − 1) to
(m, n, o , p)with rate (n+ 1)Mvδ20∗ .
The balance equations for this Markov process are ex-pressed as δ2∗+ δ4∗+ δ∗18nMv+ δ∗9+ δ∗11"v+ δ2∗mηv +δ6∗+ δ8∗+ δ17∗pµp+ δ13∗ + δ15∗λp πm,n,o,p∗ = δ∗1"vπm∗−1,n,o,p+ δ∗3"vπm,n∗ −1,o,p + δ5∗λpπm,n,o∗ −1,p+ δ7∗λpπm,n,o,p∗ −1 + δ∗ 10 nMv+ (m + 1)ηv πm∗+1,n,o,p + δ∗ 12(n+ 1)Mvπm,n∗ +1,o,p+ δ∗14pµpπm,n,o∗ +1,p + δ∗ 16(p+ 1)µpπm,n,o,p∗ +1 + δ19∗(p+ 1)µpπm∗+1,n−1,o,p+1 + δ∗ 20(n+ 1)Mvπm,n∗ +1,o+1,p−1, (43) where δ∗1, δ2∗, δ3∗, . . . , δ∗20 are defined in (23), (24), (25), . . . , (42), respectively. We derive Pbv and Pbp as follows. A new
voice call is blocked if one of the following two events occurs: (E1) When the voice call arrives at the cell, there are no free channels, and the VQ is full (i.e., n+ p = C and
m= V ). Then
Pr(E1) occurs =
(m,n,o,p)∈{(a,b,c,d) | a=V, 0cP
b+d=C, (a,b,c,d)∈S2}
πm,n,o,p∗ ; (44)
(E2) The voice call in the VQ leaves the cell before it is served. From [12], Pr[(E2) occurs] is expressed as Pr(E2) occurs
=
(m,n,o,p)∈{(a,b,c,d) | 0aV, 0cP,
b+d=C, (a,b,c,d)∈S2}
(m+ 1)ηvπm,n,o,p∗
nMv+ pµp+ (m + 1)ηv
.
(45)
Pbv can be obtained by summation of (44) and (45). For a
packet arrival to a cell, if no free channel is available, and the PQ is full (i.e., n+ p = C and o = P ), then this packet is dropped. The Pbp can be expressed as
Pbp =
(m,n,o,p)∈{(a,b,c,d) | 0aV, c=P,
b+d=C, (a,b,c,d)∈S2}
πm,n,o,p∗ . (46)
With (20)–(22), (44), (45), (43), and (46), we use the it-erative algorithm in [13] to compute λvh, πm,n,o,p∗ , Pncv and Pbp.
Table 1
Comparison of the analytic data of DRAVP0 and the simulation results for DRAVP (µp= 10µv, C= 7).
λp(units: µv) Analytic Simulation Error
Pbp(P= 3) 50 6.1752% 6.1922% 0.2745%
Pbp(P= 3) 100 34.4477% 34.4666% 5.5· 10−4
Pbp(P= 7) 50 1.44271% 1.4948% 3.5%
Pbp(P= 7) 100 30.9598% 30.8461% 0.37%
Table 2
Comparison of the analytic data of DRAVP1 and the simulation results for DRAVP (λv= 4µv, µp= 100µv, ηv = 0.2µv, C= 7, V = 4, P = 3,
K= 1).
λp(units: µv) Analytic Simulation Error
Pncv 25 1.23487% 1.203% 2.6% Pbp 25 8.85648% 8.74934% 1.21% Pncv 50 1.24146% 1.2359% 0.45% Pbp 50 10.8194% 10.8606% 0.38% Pncv 75 1.24792% 1.2343% 1.1% Pbp 75 12.3118% 12.3931% 0.66% 3.3. Simulation validation
We have developed a simulation model for the DRAVP algo-rithm, which is similar to that in [12]. The analytic models and the simulation experiments are validated against to each other in two parts.
Part 1. Validation of the dynamic channel allocation and buffering mechanism for GPRS packet data. In this part, we consider the packet traffic by ignoring voice call arrivals. Ta-ble 1 lists the Pbp values for the simulation experiments and
the analytic data of the analytic model DRAVP0. The details of the parameter setup in this table will be described in the following section. In this table, the errors between the sim-ulation experiments and analytic data are within 1% in most cases and are always less than 4%.
Part 2. Validation for the case when K = 1. In this part,
we consider a special case for DRAVP when K = 1. We list the Pncv and Pbpvalues of the simulation experiments and
the analytic results of the analytic model DRAVP1 in table 2, which indicates that the errors are within 2% in all cases.
For DRAVP, the analytic results and simulation experi-ments are also consistent, and the results will not be pre-sented.
4. Performance evaluation
Base on the analytic and simulation models developed in the previous section, we investigate the performance of the chan-nel allocation algorithms DRAVP and DRAPV. In this sec-tion, the input parameters λv, λp, ηv and µpare normalized by µv. For example, if the expected voice call holding time is 1/µv= 3 minutes, then λv= µvmeans that the expected
(a) (b)
Figure 4. Effects of the PQ on the voice calls (λv= 5µv, ηv= 0.2µv, µp= 100µv, C= 7, P = 7, V = 7). (a) Pncv. (b) Wv.
(a) (b)
Figure 5. Effects of the PQ on the packets (λv= 5µv, ηv= 0.2µv, µp= 100µv, C= 7, P = 7, V = 7). (a) Pbp. (b) Wp(DRAVP only).
voice call inter-arrival time at a cell is 3 minutes. Our experi-ments consider one frequency carrier (or 7 channels) per cell, that is, C = 7. Similar results are observed for various C values and will not be presented in this paper.
4.1. Effects of the packet queue PQ
In [12], we studied the performance for DRAQ_NH where the BSS only maintains the VQ for the voice calls but no PQ exists for the packet data. The results in [12] showed that DRAQ_NH effectively increases the packet acceptance rate, and its queuing mechanism for voice calls significantly reduces the voice call incompletion probability. In this pa-per, DRAVP is compared with DRAQ_NH to investigate how the buffering mechanism for the packets in DRAVP affects the performance for both voice calls and packets. Note that DRAVP is the same as DRAQ_NH, except that a PQ exists for the packets, and the requests in the VQ have higher prior-ity over that in the PQ.
Effects of the PQ on voice calls. Figure 4 shows Pncv and Wv values against λp and K for DRAVP and DRAQ_NH, where λv= 5µv, µp= 100µv, ηv = 0.2µv, C = 7, V = 7, and P = 7. Figure 4 indicates that when K and λpincreases,
Pncv and Wv values for DRAVP and DRAQ_NH are almost
identical, which is due to the fact that both algorithms give higher priority to the buffered voice calls. Therefore the intro-duction of the PQ in DRAVP does not affect the performance of voice calls (i.e., Pncv and Wv).
Effects of the PQ on the packet dropping probability (Pbp).
Figure 5(a) shows Pbp and Wp values against λp and K for
DRAVP and DRAQ_NH, where the input parameter setup is the same as that in figure 4. We observe that DRAVP outper-forms DRAQ_NH in terms of the packet dropping probability
Pbp, which impliess that the buffering mechanism for packet
requests significantly reduces the packet dropping probability. Furthermore, we observe that in DRAVP, Pbp is not affected
by the change of K. On the other hand, in DRAQ_NH, Pbpis
an increasing function of K. When K increases, the packet re-quests become more bursty. It is more likely that these packet requests find no available channels and are therefore dropped. In DRAVP, for the packet requests that cannot be served im-mediately, they can be buffered in the PQ, and have the second chance of being served.
Effects of the PQ on the buffered packet request waiting time (Wp). Figure 5(b) plots Wpas functions of K and λp. The
(a) (b)
Figure 6. The comparison between DRAVP and DRAPV (λv= 4µv, µp= 100µv, ηv= 0.2µv, K= 4, C = 7, V = 4, P = 3). (a) Pncvand Pbp. (b) Wv
and Wp.
figure shows an abnormal phenomenon that Wpdecreases as
λp increases. In this figure, µp = 100µv, λv = 5µv, and
λp 50µp, which implies that during the packet transmission period tcp:
(P.1) voice calls are not likely to arrive (only average λv·
E[tcp] = λv/µp= 0.05 voice calls arrive); but
(P.2) instead, packet requests are likely to arrive at the cell (average λpE[tcp] = λp/µp 0.5 packets arrive). As λpincreases, (P.2) is more likely to occur. If there are channels released by the voice calls during the transmission of packets, these channels are more likely to be occupied by packets as λpbecomes larger. Therefore, more channels serve for packets, and the buffered packet requests spend shorter waiting times in the PQ.
4.2. Comparison for the DRAVP and DRAPV algorithms
Figure 6 plots Pncv, Pbp, Wv, and Wp as functions of λp
for DRAVP and DRAPV. In this figure, we set λv = 4µv,
µp = 100µv, ηv = 0.2µv, K = 4, C = 7, V = 4, and
P = 3. For various input parameter setups, we observe the
same results that will not be presented in this paper. This fig-ure shows that DRAVP outperforms DRAPV in terms of the
Pncv and Wvperformance, and DRAPV outperforms DRAVP
in terms of the Pbp and Wpperformance (due to the different
priority order of the buffered packets and voice calls). When
λp is small, the improvements of Pncv and Wv for DRAVP
over DRAPV are insignificant. As λpincreases, the improve-ments become significant, which implies that with small λp, DRAPV is suitable for channel allocation. When λpbecomes large, to maintain both QoS for voice and packet data users, DRAVP is the better choice.
4.3. Effects of the variance of voice user cell residence times
We assume that the cell residence times are gamma distribu-tion with mean 1/ηvand variance vv. The gamma distribution was adopted to model the mobile user movement in many studies [4,10,11]. Figure 7 plots Pncv, Pbp, Wv and Wp as
functions of vvfor DRAVP, where the input parameter setup is the same as that in figure 6. The distributions for other input parameters are exponential. This figure indicates that:
• Pncv is an increasing function of vv.
• Pbp, Wv, and Wpdecreases as vvincreases.
The above results indicate that with a larger vv, more short cell residence times for voice users are observed. Thus the voice calls are more likely to handoff to another cell, and voice handoff traffic becomes more bursty. Consequently, voice calls become less likely to be completed. On the other hand, packets have better chance to be accepted in this case. Furthermore, shorter cell residence times lead shorter voice call channel occupancy times, and both buffered voice call requests and packets spend less time waiting in the queue.
4.4. Effects of Pareto packet inter arrival/transmission times
The Pareto distribution is widely used to approximate the WWW packet traffic very well [3]. In this paper, we investi-gate the effects of Pareto distribution. Assume that the packet inter-arrival times and transmission times are Pareto distri-bution with two parameters β and l, where β describes the “heaviness” of the tail of the distribution. The Pareto density function is fP(t)= (β/l)(l/t)β+1and the expected value is
E[t] = β β− 1 l. (47)
(a) (b)
Figure 7. Effects of the variance of voice user cell residence times (λv =
4µv, λp= 25µv, µp= 100µv, K= 4, C = 7, V = 4, P = 3). (a) Pncv
and Pbp. (b) Wvand Wp.
Table 3
Effects of Pareto packet inter-arrival and transmission times on DRAVP (β= 1.2, λv= 4µv, λp= 25µp, µp= 100µv, νv= 0, K = 4, C = 7, V = 4, P = 3). (Arrival, transmission) Pncv(%) Pbp(%) Wv(10−2/µv) Wp(10−2/µv) 1. (exp., exp.) 1.2017 8.717 2.97251 1.41129 2. (Pareto, exp.) 1.2363 10.1734 3.06845 1.00832 3. (exp., Pareto) 1.4201 9.70191 3.38182 1.5528
If β is between 1 and 2, then the variance for the distribu-tion becomes infinity. Once a suitable value for β is selected to describe the traffic characteristics, then l is determined by the mean of the distribution. We select β = 1.2 for packet inter-arrival times and transmission times as in [3]. By sub-stituting β into (47), we obtain l= 1/(6λp)for packet inter-arrival times and l = 1/(6µp)(l = 1/(6kµp)) for transmis-sion times if one channel (k channels) allocated to the packet. Table 3 compares Pncv, Pbp, Wv, and Wpfor DRAVP in three
scenarios:
Scenario 1. Both packet inter-arrival and transmission times
have exponential distributions.
Scenario 2. Packet inter-arrival times have Pareto
distribu-tion, and packet transmission times have exponential dis-tribution.
Scenario 3. Packet inter-arrival times have exponential
tribution, and packet transmission times have Pareto dis-tribution.
The table indicates that in most cases, the three scenar-ios show similar performance (i.e., Pncv, Pbp, Wv, and Wp).
The exponential packet inter-arrival and transmission times can provide performance trend in the real world.
5. Conclusion
In this paper, we proposed analytic and simulation models to investigate the impact of the buffering mechanism on the GPRS/GSM performance. We considered two channel al-location algorithms DRAVP and DRAPV where the voice queue and packet queue are used to buffer the GPRS packet
and GSM voice call requests, respectively. In DRAVP, the buffered voice calls have higher priority to be served than the buffered packets. On the other hand, in DRAVP, the buffered packets have higher priority over the buffered calls. Our study indicated that the buffering mechanism for the GPRS pack-ets effectively increases the GPRS packet acceptance rate at the cost of slightly degrading the performance of GSM voice calls. Furthermore, our study indicated that when the packet arrival rate is small, DRAPV is suitable for channel alloca-tion. When the packet arrival rate becomes large, to main-tain QoS of both GSM voice calls and GPRS packet services, DRAPV is the better choice.
Acknowledgements
P. Lin’s work was sponsored in part by the NSC, ROC, under the contract numbers 2213-E-002-065 and NSC91-2219-E-002-048, FarEastone, CCL, ITRI, ROC, and Chung-Shan Institute of Sci. & Tech., ROC.
Appendix. Input parameters and output measures
The output measures evaluated in our study include: • Pbp: the GPRS packet dropping probability.
• Pncv: the GSM voice call incompletion probability (i.e.,
the probability that a voice call is blocked as a new call attempt or forced to terminate as a handoff call attempt). • Wp: the average GPRS packet waiting time.
• Wv: the average GSM voice call waiting time. The input parameters set up in our study are:
• β: the parameter to describe the heaviness of Pareto packet inter-arrival (transmission) times.
• λp: the GPRS packet request arrival rate to a cell. • λv: the GSM voice call arrival rate to a cell.
• µp (kµp): the transmission rate when a single channel (k channels) is (are) used to deliver a GPRS packet re-quest.
• 1/µv: the mean of GSM voice call holding times. • 1/ηv: the mean of cell residence times of a GSM voice
user.
• vv: the variance of the Gamma GSM voice user cell resi-dence times.
• C: the total number of channels in a cell.
• K: the number of channels specified in a GPRS packet request.
• P : the maximum number of packet requests that could be buffered in the PQ.
• Q: the maximum number of GSM voice call requests that could be buffered in the VQ.
References
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Phone Lin received his B.S. degree and Ph.D. de-gree from National Chiao Tung University, Taiwan, ROC in 1996 and 2001, respectively. In 2001, he was appointed as an assistant professor of Depart-ment of Computer Science and Information Engi-neering (CSIE), National Taiwan University, ROC. His current research interests include personal com-munications services, wireless Internet, and perfor-mance modeling.
E-mail: [email protected] WWW: http://www.csie.ntu.edu.tw/˜plin
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