Analysis of a Hierarchical Cellular System with
Reneging and Dropping for Waiting
New and Handoff Calls
Cooper Chang,
Student Member, IEEE,Chung-Ju Chang,
Senior Member, IEEE,and Kuen-Rong Lo
Abstract— In this paper, we analyze a hierarchical cellular
system with finite queues for new and handoff calls. Both the effect of the reneging of waiting new calls because of the callers’ impatience and the effect of the dropping of queued handoff calls as the callers move out of the handoff area are considered, besides the effect of guard channel scheme. We successfully solve the system by adopting the multidimensional Markovian chain and using the transition-probability matrix and the signal-flow graph to obtain the average new-call blocking probability, the forced termination probability, and the average waiting time of queued new and handoff calls. We further investigate how the design parameters of buffer sizes and guard channel numbers in macrocell and microcells affect the performance of the hierarchi-cal cellular system. The results show that provision of buffering scheme and guard channel scheme can effectively reduce the new-call blocking probability and the forced termination probability in the hierarchical cellular system, and the effectiveness is more significant in the macrocell than in the microcells.
Index Terms— Buffering scheme, dropping, guard channel
scheme, hierarchical cellular system, reneging.
I. INTRODUCTION
O
NE OF THE important engineering issues in cellular communication systems is to improve spectrum effi-ciency because teletraffic demands for wireless communica-tions services are increasing. Microcell systems can be given more channels per unit coverage area than macrocell systems so that the spectrum efficiency of microcell systems is better than that of macrocell systems. However, microcell systems are not cost effective in areas with low-user population density due to base-station building cost; they are also not suitable for high-mobility users with large handoff rate. Therefore, cellular systems with hierarchical structure were proposed to take advantages of both microcell and macrocell systems [1]–[5].Rappaport et al. proposed an early personal communica-tion network with a radio link architecture that combines both centralized and distributed control to provide a low-cost mobile radiotelephone service [1]. In [2], Steele and Nofal proposed a priority handoff scheme in a hierarchical personal communication system to reduce the forced ter-mination probability of calls. I et al. studied the spectrum sharing strategies in terms of maximizing network capacity Manuscript received May 7, 1997; revised October 9, 1997. This work was supported in part by the National Science Council, Taiwan, R.O.C., under NSC Contract 85-2213-E-009-006.
The authors are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C.
Publisher Item Identifier S 0018-9545(99)05746-1.
in a hierarchical architecture [3]. Yeung and Nanda pro-posed a macrocell/microcell selection strategy, which can dynamically adjusted the velocity threshold, for a two-tier microcell/macrocell cellular system to increase the traffic load of the system [4]. Rappaport and Hu proposed an overflow scheme for hierarchically communication systems to reduce the blocking probabilities of both new and handoff calls [5]. All the above studies did not consider the buffer provision for the hierarchical cellular system.
Blocking probabilities of new and handoff calls are impor-tant performance indexes in designing cellular communication systems. Buffer provisioning for new and/or handoff calls can reduce the blocking probability of new and/or handoff call attempts [6]–[11]. Queueing for handoff calls is necessary since the terminal spends time, named handoff-dwell time, in the handoff area. Queueing for new calls is possible due to the patience of users. Gu´erin considered a system with infinite new-call buffer size, but neglected the reneging of the queued new call [8]. Hong and Rappaport proposed an appropriate analytical model and derived performance measures for a cellular mobile telephone system with infinite queueing of handoff calls [7], [9]. We studied a system with finite queues for both new and handoff calls and took reneging and dropping processes into consideration [10]. Because blocking of new-call attempts is more tolerable than forced termination of ongoing calls, guard channel scheme for handoff attempts was also proposed to minimize the blocking probability of handoff calls. Zeng et al. considered a system with finite queues for both new and handoff calls, but neglected reneging of waiting new calls [11].
In this paper, we analyze a hierarchical cellular system with finite queues for both new and handoff calls. Overlaid microcells cover high-teletraffic areas to enhance system ca-pacity. Overlaying macrocells cover all of the territory to provide general service in low-teletraffic areas and to provide channels for calls overflowing from the overlaid microcells. Guard channels and waiting queues are provided for handoff calls to minimize the forced termination probabilities; waiting queues are also supported for new calls to reduce the new call blocking probability. Buffered handoff calls are given service priority higher than buffered new calls because interruption of ongoing calls upset customers more than blocking of new calls. Reneging process of waiting new calls and dropping process of waiting handoff calls are further considered in this paper.
Fig. 1. A typical macrocell: N microcells and an overlaying macrocell. Analysis is via a multidimensional Markov chain approach. The state probabilities are obtained by using state-transition equations since the system possesses a quasi-birth–death Markovian property [12]. We successfully derive the reneging probability of waiting new calls by using a transition-probability matrix approach. The transition-transition-probability matrix is composed of one-step state-transition probability of the system, which can be used to find multistep state-transition probability by matrix multiplication. Via some substitution and derivation, the transition-probability matrix approach can fur-ther be used to obtain the average waiting time of queued new calls, and the performance measures of handoff calls are also obtained by signal-flow-graph approach. Moreover, we heuris-tically define a cost function to investigate the optimal guard channel pattern of the system and the suitable queue sizes for both new and handoff calls in the hierarchical cellular system. The rest of the paper is organized as follows. The channel assignment strategy and basic assumptions for a hierarchical cellular system are presented in Section II. In Section III, we derive the new-call blocking probability and waiting time, the forced termination probability, and the handoff waiting time by the transition-probability matrix approach and the signal-flow graph approach. In Section IV, an example of hierarchically overlaid cellular system with new and handoff queues is illustrated and discussed. Finally, concluding remarks are given in Section V.
II. SYSTEM MODEL
The hierarchical cellular system is assumed to consist of macrocells, in which a typical macrocell has an overlaying macrocell and its overlaid microcells. As Fig. 1 shows, the overlaying macrocell, denoted by macrocell 0, has the cov-erage area bounded by the outermost closed contour. The
overlaid microcell, denoted by microcell , has
its own coverage area. The area outside microcells, named as the macrocell-only region, is served only by macrocell 0.
The system model of a typical macrocell is shown in Fig. 2.
The th cell, , supports channels and reserves
of exclusively for handoff requests of ongoing calls from other cells. Cell also provides two waiting queues with capacities and for new calls and handoff calls, respectively.
The channel assignment scheme of the macrocell is de-scribed as follows.
1) For a new call originating in the macrocell-only region, it will be served immediately by macrocell 0 if at its arrival the number of idle channels in macrocell 0 is larger than the number of guard channels If the number of idle channels is not greater than and the new-call queue in cell 0 is not full, the call will be put in the queue. Otherwise, it will be blocked.
2) For a new call originating in the overlaid microcell , it will be served immediately by microcell if at its arrival the number of idle channels in microcell is greater than the number of guard channels If the new call cannot be served by microcell , but the number of idle channels in macrocell 0 is greater than , it will overflow to and be served by macrocell 0. If neither microcell nor macrocell 0 can serve the new call, but the new-call queue of microcell still has waiting rooms, the call will be buffered in the queue. Otherwise, the call will be blocked.
3) For a handoff call coming from neighboring macrocell because of its high-mobility behavior, it will be directed to macrocell 0 no matter in which cell the call is in the neighboring macrocell. The call will be served immediately by macrocell 0 if there are free channels in macrocell 0. If macrocell 0 has no idle channels, but has free waiting room, the call will be buffered in the queue; otherwise, the call will be blocked.
4) For a handoff call moving from microcell to macrocell-only region, it will be directed to macrocell 0. The
Fig. 2. The system model for a macrocell.
channel assignment scheme is the same as that for the handoff call in 3).
5) For a handoff call moving from microcell to its neighboring microcell , it will be served immediately by microcell if at its arrival there are at least one free channels in microcell If microcell has no idle channel, but the overlaying macrocell 0 has some free ones, the handoff call will overflow to and be served by macrocell 0; otherwise, it will be put in the queue if the queue is not full or be blocked if the queue is full. 6) In each cell, the service priority for buffered handoff
calls is higher than that for buffered new calls. That is, whenever a channel is released and becomes available
in cell , the handoff call buffered in the
head of the line in cell has the first priority to use the idle channel; the buffered new call would not be served until the number of idle channels in cell is greater than the number of guard channels.
7) For system operation simplicity, the overflow scheme is not applied for buffered new and handoff calls in microcells.
8) The waiting new call in the queue may renege from the system if it cannot access a free channel within its patience time.
9) The waiting handoff call in the queue may be dropped by the system if it cannot access a free channel within its handoff-dwell time.
10) When a mobile platform holding a channel of overlay-ing macrocell 0 moves across any one of the microcells, no handoff action is needed to take.
Other basic assumptions involved in the model are stated below. The arrival process of new calls originated in the
macrocell-only region or microcell is a Poisson
process with mean rate or The arrival process of handoff calls from neighboring macrocell is a Poisson process with mean rate The unencumbered session duration of a call, denoted by , is exponentially distributed with mean The time spent in a cell by a mobile is named by cell-dwell time. The cell-dwell time associated with new and handoff calls is assumed to be the same here. The cell-dwell time in cell , denoted by , is exponentially distributed with mean The patience time of waiting new calls in cell , denoted by , is exponentially distributed with mean The handoff-dwell time of waiting handoff call to cell , denoted by , has an exponential distribution with mean The average fraction of handoff departure from
cell to cell is denoted by , where denotes the
macrocell-only region, denotes the microcell,
and denotes the neighboring macrocell. Clearly,
if and The system is
assumed to be homogeneous with the same assumption listed above for all macrocells.
III. ANALYSIS A. The System-State Probabilities
We define the system state for the macrocell as , where denotes the sum of the number of communicating platforms and waiting handoff
calls, and denotes the number of waiting new calls, The states form a -dimensional sample space which is given by
(1) We can obtain the limiting probability of state denoted by by solving the following stationary state-transition equations:
for all (2)
and the probability conservation condition
(3)
where denotes the transition rate from state to
state and denotes the transition rate out of state can be obtained by
(4)
In the following, we determine the transition rate
from state to state based
on the assumptions and statements 1)–10) of the channel assignment scheme adopted in the previous section.
New Call Arrival:
1) Denote to be the transition rate from state to state as a new call originates in the macrocell-only region. If macrocell 0 has free channels for the new call, i.e.,
, the call would be served immediately by macrocell 0 and the transition rate is
If macrocell 0 has no available channel for the new call, but has waiting rooms in the new-call queue, i.e.,
, the new call would be buffered
in the new-call queue and Otherwise,
2) Denote to be the transition rate from state to state as a new call originates in microcell
If
, the call would be served
immediately by cell and If
, the call would overflow to and be served immediately by macocell
0 and If
, the call would be buffered in the new-call queue of cell and
Otherwise,
Handoff Call Arrival:
3) Denote to be the transition rate from state to state as a handoff call comes from its neighboring macrocell. If macrocell 0 has free channels or has no available channel, but has waiting rooms for the handoff calls, i.e.,
or
the handoff call will be immediately served or queued by
macrocell 0 and Otherwise,
4) Denote to be the transition rate from state to state as a handoff call arrives at the macrocell-only
region from microcell If case A
or or case B
the handoff call would be immediately served or queued by macrocell 0 in case A, or be blocked by the system in case B. The transition rate in case A or case B would
be for or or for If case C and or or case D
the handoff call would be served or queued by macrocell 0 in case C and be blocked by the system in case D. The transition rate in cases C or D would be
for Otherwise,
5) Denote to be the transition rate from state to state as a handoff call arrives at microcell
from microcell If case A
or case B or case C or case D
the handoff call would be served immediately in microcell in case A, be overflowed to and be served by macrocell 0 in case B, be queued in microcell in case C, and be blocked by the system in case D. The transition rate in all these cases is
for or for If case E, or case F or case G or case H
the call would be served immediately by microcell in case E, be overflowed and be served by macrocell 0 in case F, be queued in microcell in case G, or be blocked by the system in case H. For all these cases,
for Otherwise,
Reneging (Dropping) of New (Handoff) Calls:
6) Denote to be the transition rate from state to state as a new call reneges from the new-call queue
of cell If
denoting that there are waiting new calls in the new-call queue of cell the transition rate would
be Otherwise,
7) Denote to be the transition rate from state to state as a handoff call is dropped from the handoff-call
queue of cell If
the transition rate would be
Otherwise,
Handoff Call Departure:
8) Denote to be the transition rate from state
to state as a handoff call departs from cell to its neighboring macrocell. If denoting that the handoff queue of cell has no waiting handoff calls as the handoff call departure from cell occurs, the transition rate would be
If
denoting that cell has waiting handoff calls as the handoff call departs from cell the transition rate would be
If
denoting that the number of available channels equals to the number of guard channels in cell as the handoff call departs from cell the transition rate would be
Otherwise,
Call Completion:
9) Denote to be the transition rate from state to
state as a call is released in cell If
the transition
rate would be for
or or
for If
the transition rate is given by Otherwise,
Then can be obtained by summing up all
possible transition rates from state to state which
can be expressed as
(5)
B. The Performance Measures
1) Average New-Call Blocking Probability: Here, blocking of new calls in the system occurs in two situations. One is that, at the arrival instant of the new call, the system has no available channel and waiting buffer; the other is that although the new call is temporarily accepted and put in the new-call queue, it finally reneges from the queue due to impatience. For a new call originating in the macrocell-only region or microcell , the set of system states that belongs to the first situation,
denoted by or can be expressed as
(6)
(7) And the set of system states at which an originating new call joins the new-call queue of macrocell 0 or microcell denoted
by or can be expressed as
(8)
(9) Thus, the blocking probability of a new call originated in macrocell-only region or microcell denoted by or , respectively, can be expressed as
(10)
where is the reneging probability of a waiting new call in cell given that the system is at state as it just arrives. We use a quasi-system state to describe the transition process of a waiting new call and a transition-probability
matrix method to obtain The derivation of is
given in the Appendix A. Therefore, the average new-call blocking probability of the system, denoted by can be intuitively obtained by
(11)
2) Average Waiting Time of Queued New Calls: Here, we only consider the average waiting time of queued new calls which can successfully access a free channel before reneging.
Let denote the average waiting time of a queued
new call which joins in cell as the system state is at The average waiting time of a queued new call
in cell denoted by , can be obtained by
(12)
And the average waiting time of the system, denoted by , can be obtained by
(13)
In the derivation of , as in the Appendix A, we use quasi-system state for cell to describe state transitions
of the queued new call. Denote to be the average
holding time at quasi-system state under the condition that the quasi-system transits to quasi-system state Clearly
(14)
where is the set of quasi-system states and is the transition rate from quasi-system state to quasi-system state
The definitions of and are given in (A.3) and
(A.13), respectively. For a queued new call which is initiated
at quasi-system state and transits steps, via
transient states to service state (the
service state is a state at which the call will be served by
the system and which is defined in the Appendix
A), the average waiting time of this transition path would
be and the transition probability would be
Therefore, can be obtained by (15), given
at the bottom of the page. Note that disappears
as The denominator in (15) is equal to
because it denotes the probability that the queued new call can be successfully served. As for the numerator, since
(16)
and the term within the square bracket of (16) is given by
substituting the matrix element in (A.12) with ,
it can be obtained. Therefore, the average waiting time in (15) is yielded.
3) Forced Termination Probability: There are three kinds of handoff attempts in the system. The first is the handoff arriving at the overlaying macrocell 0 from adjacent macro-cells, the second is the handoff arriving at the macrocell-only region from microcell , and the third is the handoff arriving at the microcell from microcell
Failure of a handoff call coming from its neighboring macrocell occurs in two situations. One is that the handoff call is blocked at its arrival, the other is that the handoff call is temporarily buffered in the waiting queue, but is dropped by the system because the mobile moves out of the handoff area. The sets of system states belonging to these two situations are denoted by and , respectively, which can be given by
(17) (18) Because the handoff arrival rate from neighboring macrocells is assumed to be uniform, the handoff failure probability, denoted by , can be obtained by
(19)
where is the dropping probability of a queued handoff call in macrocell 0 given that the system is at state as it just arrives. We also use quasi-system state to describe the transition process of a queued handoff call and the signal-flow-graph method to obtain The derivation of is given in the Appendix B.
Failure of a handoff call coming from microcell to macrocell-only region occurs also due to being blocked imme-diately or being dropped from the queue. Because the handoff arrival rate is proportional to the number of communicating
users in microcell , we define Prob system is in
state a handoff attempt from microcell occurs can
be expressed as
(20)
where
if
if (21)
Thus, the handoff failure probability from microcell to macrocell-only region, denoted by , can be obtained by
(22)
Handoff failure of a call coming from microcell to microcell occurs also in two situations. They are being blocked immediately and being dropped from the queue. The sets of system states belonging these cases are denoted by and , respectively, which can be given by
(23)
(24) Because the handoff arrival rate is proportional to the number of communicating users in microcell , the handoff failure probability, denoted by , can be expressed as
(25)
Once a call is served by macrocell 0, the call will handoff only to neighboring macrocells with a handoff requirement probability, denoted by is defined as and given by
(26) And the forced termination of an arbitrarily selected call originated in macrocell-only region, denoted by , can be obtained by
(27) Similarly, the handoff requirement probability of the call
served in microcell , denoted by , is defined
as and given by
(28)
And the forced termination of an arbitrarily selected call originated in microcell , denoted by , can be obtained by
(29)
where is the overflowing probability for a handoff attempt from microcell to microcell The overflowing probability
can be obtained by
(30)
where is defined as the set of system states at which the handoff attempts from microcell to microcell will overflow to macrocell 0. can be expressed as
(31) can therefore be obtained by solving the set of linear equations shown in (29).
The forced termination probability of the system, denoted by , can be obtained by averaging over the forced termination probability of a call which may originate in the macrocell-only region or microcell of the system. is given by (32) at the bottom of the page, where is the overflowing probability of a new call originated in microcell can be expressed as (33)
where is defined as the set of the system states at which the new call attempts originated in microcell can overflow to and be successfully served by macrocell 0. can be expressed as (34) 4) Average Waiting Time of Queued Handoff Calls: Similar to the case of new calls, we only consider the average waiting time of queued handoff calls which can successfully access a free channel before moving away from the handoff area. Let denote the average waiting time of a queued handoff
call which joins in cell at the instant when the
system state is at can be obtained by
(35)
Let denote the handoff arrival rate impinging on cell at the instant when the system state is at is given by
if
if
(36)
The average waiting time for handoff calls of the system, denoted by can therefore be obtained by
(37)
IV. NUMERICAL EXAMPLES ANDDISCUSSIONS In the following examples, a typical macrocell in a hierarchical cellular system is considered to contain one overlaying macrocell and two overlaid microcells. We assume the following system parameters:
s, s,
s, s,
and
The new-call arrival rate in each cell is set to
be the same: We use an iterative method to
numerically compute the state probability of the system and show the effects of guard channels and queue capacities of new and handoff calls on the system performance measures.
Fig. 3 shows the average new-call blocking probability and the forced termination probability versus the new-call arrival rate of the system for various guard-channel patterns, where
and It can be seen that
as increases, increases and decreases, while as
and increase, both and increase. Usually, the guard channel scheme for handoff protection would improve the forced termination probability, but deteriorate the new-call blocking probability. The reason why both and deteriorate as and increase is due to the overflow scheme of the system. The overflow scheme provides a new or handoff call impinging on a microcell an alternative of being served by macrocell 0 as the microcell has no free channel to access. The probabilities of new-call blocking and handoff failure in macrocell 0 are consequently much larger than these probabilities in microcells, and therefore the former probabilities play dominant roles in and Also note that the increment of guard channels in microcells would
Fig. 3. The probabilitiesPBn andPF versus new-call arrival raten for various guard channel pattern(Ch0; Ch1; Ch2):
Fig. 4. The cost functionB versus the number of guard channel for various
Ch0 or Ch1 (Ch2):
enforce the overflow mechanism and consequently induces much deterioration on and Therefore, in a hierarchical cellular system with overflow scheme, provision of guard channel scheme in microcells will deteriorate both new-call blocking probability and the forced termination probability of the system.
Because interruption of handoff calls upset customers much more than blocking of new calls, we heuristically define a cost function of overall blocking probability, denoted by to evaluate the quality-of-service of the system. is given by
(38) where is a weighting factor to express the stress of and
laid on the quality-of-service, Fig. 4 shows
the cost function versus as (depicted
by the solid line), and versus and
as (depicted by the dotted line), for , given
that calls per second. We observe that the dotted
line increases monotonously as and grow up, but
the solid line has a minimum value at This result tells us the macrocell needs reservation of guard channels for handoff protection, and the optimal guard-channel pattern for the hierarchical cellular system is is (2, 0, 0), with
Fig. 5 shows the probabilities and versus
Fig. 5. The probabilitiesPBn and PF versus new-call arrival raten for various queue-size patterns of new calls(Nn0; Nn1; Nn2):
Fig. 6. The cost functionB versus queue sizes for new call Nn0orNn1
(Nn2):
We find that the increment of new-call queue sizes in each cell induces improvement on , but deterioration on and the improvement of is more significant than the deterioration of for all traffic loads. Both the improvement on and the deterioration on become saturated as the queue capacities become larger because of the effects of the reneging of queued new calls and the dropping of queued handoff calls.
The change of and caused by is more significant
than by and It is owing to the effect of the overflow scheme used in the system. Fig. 6 shows the cost function
versus as (depicted by the solid
line) and versus and as
(depicted by the dotted line) with and
calls per second. It can also be found that plays a dominant effect over of microcell in the hierarchical cellular system with overflow scheme. The saturation of caused by queue size of macrocell 0 or microcells can be clearly observed from the figure, and of queue-size pattern
is sufficient in this example.
Fig. 7 shows the probabilities and versus
for different queue-size patterns of handoff calls, where As the size of handoff queue in each cell increases, decreases, but remains almost unchanged for all traffic loads. It is because the size of handoff queue has a direct effect on performance of handoff calls, but an indirect effect
Fig. 7. The probabilitiesPBn andPF versus new-call arrival raten for various queue-size patterns of handoff calls(Nh0; Nh1; Nh2):
Fig. 8. The cost functionB versus queue sizes for handoff call Nh0 or
Nh1 (Nh2):
on that of new calls. The increments of and have
almost no effect on the performance measures. It is due to the overflow scheme of the system as mentioned before. The saturation of and as the queue sizes of handoff calls increase can also be observed in the figure. The phenomenon results from the reneging and the dropping of queued new and handoff calls. Fig. 8 shows the cost function versus
as (depicted by the solid line) and
versus and as (depicted
by the dotted line), with and calls per
second. The saturation of caused by increasing buffer sizes can also be observed clearly from this figure, and
is appropriate.
Fig. 9 shows the average waiting time of queued new calls, , and the waiting time of queued handoff calls, , versus
with appropriate design parameters: and
We can see from the figure that as the new call arrival rate increases, the increment in is more significant than the increment in This is because waiting handoff calls have higher priority than waiting new calls.
V. CONCLUDING REMARKS
In this paper, we successfully analyze a hierarchical cellular system with overflow scheme, where the system supports finite
Fig. 9. The average waiting timesWnandWhversus new call arrival rate
n:
buffers for new and handoff calls. The new call may renege from the system, while the handoff call may be dropped by the system. The guard channel scheme for handoff protection is also considered. We obtain the new-call blocking probability by using transition-probability matrix approach, the forced termination probability via signal-flow graph, and then the average waiting times of new and handoff calls. Interesting system phenomena showed that because of the provision of overflow scheme, design parameters of guard channel and queue size in microcells could be negligible, while these parameters in overlaying macrocell are significant. In other words, in a hierarchical cellular system with overflow scheme, it seems more significant to support guard channel for handoff protection and buffers for new and handoff calls in overlaying macrocell than to provide them in the microcells. We further heuristically propose a cost function not only to justify the above statement, but also to determine the optimal guard channel patterns and the appropriate queue-size patterns for the hierarchical cellular system.
In the study of this paper, all of the channels in the overlaying macrocell are used by the overflowing calls from microcells; the overflow scheme would induce higher traffic load in the overlaying macrocell and thus deteriorate the per-formance in the macrocell. Therefore, it seems appropriate to set a threshold to confine the maximum number of overflowing new/handoff calls for balancing traffic loads in macrocell and microcells. The topic is now under study.
APPENDIX A DERIVATION OF
In order to obtain , we use quasi-system states for cell to describe state transitions of the waiting new call in cell The quasi-system states for cell are composed of two kinds of states: absorbing states and transient states. The absorbing states are either the states at which the waiting new call in cell will renege from the system (the state is called the reneging state) or the states at which the call will be served by the system (the state is called the service state). The transient states are the states denoting that the waiting new call is still in the queue. Except for the reneging state, the quasi-system states for cell are defined as the system states excluding those
new-call arrivals in cell queueing behind the waiting new new-call. The reneging state is a virtual state; we here denote it by Then the set of absorbing states, denoted by , can be expressed as
with constraint of
(A.1) The set of transient states, denoted by can be obtained by
with constraint of
(A.2) And the set of the quasi-system states, denoted by , is given by
(A.3) can be obtained by summing up the transition prob-abilities of all possible transitions from state to reneging state where state is a quasi-system state in Denote to be the -step transition probability from quasi-system
state to quasi-system state where Based
on the memoryless assumptions of the system, the -step quasi-system state-transition probability can be obtained by
(A.4)
where denotes one-step transition probability from quasi-system state to quasi-system state In matrix form, we have
(A.5)
where is the transition-probability matrix ,
and is the th power of
Without loss of generality, can be expressed as
(A.6)
where denotes the transition-probability matrix
from states in to another states in Because all states in are absorbing states and the transition from an absorb-ing state to any other states is impossible except to itself,
must be an identity matrix and must
be a zero matrix. So becomes
(A.7) Then, can also be expressed as
(A.8)
where and can be obtained by
and
(A.10) Equation (A.9) shows that the -step transition probabilities from states in to states in are composed of all possible
transition probabilities via exactly steps: A
-step transition consists of exactly steps of transition among the transient states and a one-step transition from the transient state to the absorbing state. Therefore, can be obtained from the transition-probability matrix by
(A.11)
where is an element in the transition matrix
which is given by
(A.12) Let denote the transition rate from quasi-system
state to quasi-system
state Based on the assumption of the system and the definition of the quasi-system, we have
if if if
elsewhere
(A.13) where is the transition rate from the system state to system state as defined in Section III. Because the driving processes of the system are assumed to be memoryless, can be obtained by
(A.14)
APPENDIX B DERIVATION OF
In order to obtain we also use quasi-system states for cell to describe state transition of the waiting handoff call which joins the queue at system state The quasi-system states for the handoff call are also composed of transient states and absorbing states. The transient states denote that the waiting handoff call is still in the queue. The absorbing states, composed of dropping state and service states, denote that the waiting handoff call of interest leaves the waiting queue. The dropping state, which is a virtual state and is denoted by is the state that the waiting handoff call is dropped from the queue as it moves away from the handoff area. The service states are the states that the call can be served by the system. For the waiting handoff call, the quasi-system states, except the dropping state can be simply identified by which is the sum of the number of busy channels in cell and waiting handoff calls in cell coming before the call of interest.
Fig. 10. The transitions of quasi-system states for handoff attempts impinge on cell i:
Fig. 10 shows a signal flow graph that portrays the transition of quasi-system states from the input state to the output states, which are the service state or the dropping state
The one-step transition rate from quasi-system state to quasi-system state denoted by can be expressed as
if if
elsewhere. (B.1) The probability that the call of interest can access a free
channel is because the call of interest must be
either served or dropped by the system. Obviously, can be obtained by
(B.2)
where the initial system state
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Cooper Chang was born in Keelung, Taiwan, R.O.C. He received the B.S. degree from National Chiao Tung University, Hsinchu, Taiwan, in 1990 and the M.E. degree in electrical engineering from National Tsing Hua University, Taiwan, in 1992. He is currently working towards the Ph.D. degree in communication engineering at National Chiao Tung University.
His current research interests include performance analysis of wireless communications systems and personal communications services.
Chung-Ju Chang (SM’94) received the B.E. and M.E. degrees in electronics engineering from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1972 and 1976, respectively, and the Ph.D. degree in electrical engineering from National Taiwan University, Taiwan, in 1985.
From 1976 to 1988, he was with the Telecom-munication Laboratories, Directorate General of Telecommunications, Ministry of Communications, Taiwan, as a Design Engineer, Supervisor, Project Manager, and then Division Director. There, he was involved in designing a digital switching system, ISDN user-network interface, and ISDN service and technology trials. He also acted as a Science and Technical Advisor for the Minister of the Ministry of Communications from 1987 to 1989. In August 1988, he joined the faculty of the Department of Communication Engineering and Center for Telecommunications Research, College of Electrical Engineering and Computer Science, National Chiao Tung University, where he is currently a Professor. He was Director of the Institute of Communication Engineering from August 1993 to July 1995.
Dr. Chang is currently serving as the Chairman of the IEEE Vehicular Technology Society, Taiwan Chapter. His research interests include performance evaluation, PCS, and broad-band networks.
Kuen-Rong Lo was born in Hsinchu, Taiwan, in 1957. He received the B.S. degree in electrical engineering from National Taiwan Institute of Tech-nology, Taipei, Taiwan, in 1984, and the M.S. degree in computer science from National Tsing Hua University, Hsinchu, Taiwan in 1989.
He is currently working toward the Ph.D. degree at National Chiao Tung University, Taiwan. He is also a Project Leader in the Telecommunication Laboratories Chunghwa Telecom Co., Ltd.. His re-search interests lie in the broad areas of mobile communication systems and integrated services digital networks. Presently, his research is focused on channel assignment schemes, traffic performance modeling, and analysis for mobile cellular communication systems.
